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-torsion-free and divisible module, so that B = to(A). Since Anno(B) * < 0 >, there exists a D-submodule E such that we may decompose A = B@ E (see, for example, [KI]). But E, being isomorphic with AIB, is also divisible, so that AP = EP = E. This implies that £ is a DG-submodule, because AP has this structure. Furthermore, if Anno{B) = < 0 >, we find out that B is D-divisible. As above, we have a D-lineal decomposition A = B © E\, for some Z)-submodule E\ (see, for example, [SV, Proposition 2.10 and Theorem 2.15]). If AnnD{AIB) *< 0 >, it follows that E\ is the D-periodic part of A, and so E\ itself is a DG-module. If AnnD{AIB) =< 0 >, AIB becomes D-divisible too and we may consider A as a A^G-module where K is the field of fractions of D, which allows to apply Theorem 4.1 again. Our next consequence will need a notion which fairly extends that of the simple module. As usual, in the conception of these ideas, it comes from the ring
42
Simple Modules
of integers Z. Let R be an integral domain and take G to be an arbitrary group. We say that the RG-module A is R-irreducible if A satisfies the two following conditions: (i) A is R-torsion-free. (ii) For every non-zero RG-submodule B of A, the factor-module AIB is R periodic. Thus, for R = Z, Z-irreducible = torsion-free rationally irreducible, the mentioned idea from which this concept comes. Obviously a simple #G-module satisfies the above condition (i), (ii). As usual, the concept can be characterized through the extension ofscalars. That is, if K is the field effractions of J?, an R-torsion-free RG-module is R-irreducible if and only if the KG-module A ®RK is simple. 4.4. Corollary Let D be a Dedekind domain, X a formation of finite groups, G an XC-hypercentral group, A be a DG - module and B a DG - submodule of A satisfying the following conditions: (i) A is a D-torsion-free module; (ii) B is D-irreducible; (Hi) G/CG(A/B) e XbutGICG(B) £ X. Then there exists a non-zero DG-submodule C ofA such that B C\C = 0. Proof Put £ - A D K, where K is the field of fractions of D and think of £ as a ATG-module. Define B\ = B £> K, so that B\ is a simple .KG-module and CGOBI) = CG(B).
Given a e A\B, put E\= aKG + B\. Since CG(EIBi) = CG(AIB), G/Ca(Ei/Bi) eX From X c T it follows that dimK(E\IB{) is finite. In particular E\IB\ includes a simple /^G-submodule, say EilB\. It readily follows that the modules Ei,B\ and EilB\ satisfy all the conditions of Theorem 4.1. Hence there exists a /TG-submodule C\ < £2 such that £2 = B\ © C\. Since E is a D-essential extension of A, C = C\ f\A * < 0 >. But B D C = < 0 >, so C is the required module. In the study of just non-X-groups it is very important to know when a group G splits over its Af-residual. Note that theX-residual of a group G is the intersection of all normal subgroups of G giving a quotient an A'-group. We recall that a group G is splittable over its normal subgroup H (or that G splits over H) if H admits a complement (as a subgroup) in G, that is, if there exists a subgroup Q such that G = HQ and H(~)Q = < 1 >. We indicate this as G = H X Q. If all the complements to H in G are conjugate (in G), we say that G conjugately splits over H. Often a monolithic just non-A'-group conjugately splits over its monolith. It could be proved for each particular class X. However, there exists a very important
Complements of Simple Submodules
43
result, which is happen to be univesal for all nedeed cases. It is the following Robinson's theorem. 4.5. Theorem [RD 22] Let G be a group with an abelian normal subgroup A satisfying the minimal condition on its G-invariant subgroups, H a normal subgroup ofG satisfying the following conditions: (/') H > A, HI A is locally nilpotent; (ii) the upper FC - hypercenter ofGICtiiA) includes HICH(A)\ (at) AH an) =< i >• Then every over group L > A ofA conjugately splits over A. We omit the proof of this theorem, since it requires a very specific technique, straying far from our goals. One can find this proof in the paper [RD 22] and also inthebookfAFdeG].
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Part II Just Infinite Modules
As we already noted, another type of modules which play a relevant role in the study of just non-^-groups are the just infinite modules. Because for some noetherian modules it is possible to make a reduction to just infinite modules, the investigation of just infinite modules is important by itself. These modules arise by the following way. Let A be an infinite noetherian module over a ring R, M. = {C\C is an R- submodule of A such that AIC is infinite} Then M has a maximal element M. Put V = AIM. If U is a non-zero submodule of V, then VIU is finite. The following two cases are possible here: (1) V includes a non-zero simple submodule S; (2) the intersection of the family of all non-zero submodules of Vis zero. In the first case S is infinite and VIS is finite; therefore, it could be reduced to the cases of infinite simple modules and finite modules. Moreover, for some types of rings V is exactly a simple module; for example, if R = ZG, where G is a hypercentral, (even FC - hypercentral, group). Hence the second case here is the main. Let R be a ring. An R - module A is said to be a just infinite if it satisfies the following conditions: (JI I) for every submodule B of A a factor - module A/B is finite; (JI2) the intersection of the family of all non-zero submodules of A is zero. These modules have been introduced by D. J. S.Robinson and J. S. Wilson in their fundamental paper [RW], where the groups with polycyclic proper factor-groups have been studied . Since a simple ZG - module over a polycyclic group G is finite ( see, for example, [PD 1, Theorem 12.3.7]), we recall that the
45
46
Just Infinite Modules
condition (JI 2) is optional for a just infinite ZG - module. The further researches have shown that these modules play an important role in the study of just non-^-groups and for many other important classes X of groups, in particular, for all classes of groups X, which will be considered in this book. Note that just infinite modules play a crucial role also in the study of the generalized soluble groups with the weak maximal and minimal conditions for normal subgroups and in some other important investigations.
Chapter 5 Some Results on Modules over Dedekind Domains
For the study of a module A over a group ring DG where D is a Dedekind domain the structure of A as a Z)-module plays a very important role. In this chapter we have collected some results about modules over Dedekind domains, which are necessary for the study of just infinite modules over DG. In particular, we consider an analogy of the important concept of a p - basic subgroup in Abelian Group Theory. We have no found analogies of this concept for modules over Dedekind domains in the journal literature. In this case, as well as in all remaining cases, we followed the analogy from Z to D. Our purpose is to prove the existence of P- basic submodules in torsion-free modules and, using this, to prove some results about torsion-free D-modules, which are necessary for us. Thus, this chapter hass a technical character. Let D be a Dedekind domain, P a maximal ideal of D. Then the D - modules DIP" and P/P"+l are D - isomorphic (see, for example the proof of Corollary 3.3.15 from [KG]). In particular, DIP" is embedded in D/P"+l, n e N. Therefore we can consider the injective limit of the family {DIP"\n e N}. Put Cr- = lim{DIP" | n&N}. module.
The D - module CP» is called a Prufer P -
It follows from the construction of C/»» that Q/> j n (0°) =D DIP" and Q / \ n + i ( C » ) / Q ; > ( C » = (DIP"+l)l(DIP") = DIP. Hence if C is a proper D submodule of 0 ° , then C = Clp,„Cp™ for some n e N. Moreover, if c e Clpt„(Cp°>)\Qpt„-i(Cp«), then C = cD. A Prufer P-module is monolithic with monolith Qpj (C/>»).
47
48
5.1. Lemma
Just Infinite Modules
A Prtifer P - module is D - divisible.
Proof Let a e O » , 0 = Q P j „(C/»). Let / = xD. If IS P, then I+P = D. In this case D = 7 + P", so that 1 = xu+y for some u e £>,>> e P". Then a = a l = axu + ay = axu = (au)x. Consider the case when x e P. Put R = (") reN Ps • If we assume that R * < 0 >, then i? = Pi... Pt for some maximal ideals Pi,... ,Pt (see, for example, [KG, Theorem 3.3.5]). Since Pi... P, < Ps for any * e N, s < t (see, for example, [KG, Lemma 3.3.4]). So, we obtain a contradiction. This proves that R = < 0 >. Hence there exists a number t such thatx e P'\P'+1. Let B = Q/>in+;(C/>«>). The mapping
=A0 + C. Now B = A/C = (Ao + C)/C = Aol(Ao fl C) is a finitely generated submodule. 5.5. Corollary Let D be a Dedekind domain, P a maximal ideal of D, A a P module over the ring D. If A is an artinian D - module, then A = a\D © ... © asD © C\ © ... © C, where C, is a Priifer P- module, 1 < i < t. Proof We can consider AIAP as a vector space over the field DIP. Since A is artinian then dimD/PAIAP is finite. Corollary 5.4 implies that^4 = B © C where B is a finitely generated submodule, C is a divisible submodule. Then B = ai£> © ... © asD (see, for example, [CUR 1, Corollary 22.16). Let 0 * c\ e C, y e P\P 2 . Since C is D - divisible, there are elements {c„ | n e N}
50
Just Infinite Modules
with the following property: c„+\y = c„, n e N. It is not hard to prove that c„D =D DIP" for each n e N. It follows that C\ = Y^„m C"D i s a P r t i f e r submodule. In particular, C\ is D - divisible by Lemma 5.1. Now we can note that every D - divisible submodule is a direct summand of the module (see, for example, [SV, Proposition 2.10 and Theorem 2.15]). 5.6. Lemma Let D be a Dedekind domain, P a maximal ideal of D, A a P -module over the ring D. Then A is an artinian module if and only if dimD/pQp,i(A) is finite. Proof If A is artinian, then Q^i (A) is artinian too. It follows that dimD/PQ.p,\(A) is finite. Lemma 1 from [ZD 5] implies the converse. 5.7. Theorem Let D be a Dedekind domain, A a D - module. Then A is artinian if and only if A = a\D © ... © asD © C\ © ... © C,, where C, is a Prufer P, module, Annoicij) = PJ,Pj,Pj are maximal ideals ofD, 1 < i < t, 1 <j < s. Proof Suppose that A is artinian. If A is not D - periodic, then A has an element a such that Anno(a) =< 0 >. Therefore aD = D. However D has an infinite descending chain D > P > P2 > ... > Ps > ... for every P e Spec{D). This proves that A is D -periodic. By Proposition 1.1, A = © p e n o u ) Ap where Ap is the P - component of A. Since A is artinian, the set TID(A) is finite. Now we can apply Corollary 5.5 to each submodule Ap. The sufficiency follows from Lemma 5.6. 5.8. Lemma Let D be a Dedekind domain, P e Spec(D), A a torsion -free Dmodule and ro(A) = 1. IfB is a finitely generated non-zero submodule of A, then either the P - component of AIB is a cyclic P - module or it is a Prufer P module. Proof Let F be the field of fractions of the ring D, 0 * b e B. If bx = by for some elements x,y e D, then 0 = bx-by = b(x-y). Since A is torsion-free, x —y = 0, i.e. x = y. Therefore if we put (bx)
z : a —• az, a e Y\. Clearly Imq>z < B. Since A is D - torsion - free, Ker, in particular, Y\ = Imq>z. Hence Y\IB = (]mfz)/{B . The action of D on F could be naturally extended to an action of Jon V. Now, VP" = lim{A/AP"+' | f € N} = V„ (see, for example, [ND, 9.10 Theorem 18]), and VIVP" = A/AP", n e N (see, for example, [ND, 9.5, Corollary of Theorem 4]). Let.y e P\P2; then P = Dy + P" for any « e N, in particular, Pm = Dym + P" if w < « by Lemma 4.2. Let (x„)„s^ e U(J); then there is an element (U„)„<EN e J such that (x„)neN(wn)«eN = 1 where 1 = (1 + P,\+P2,... ,\+P",...). In particular, 5c\u\ = 1 + P, so that it] * 0. Conversely, let (3C„)„SN e J and ici * 0. Then 3c i = x\ + P * />, that is x\ £ P. Since P is a maximal ideal of D then P + x\D = D. Let 3c„ = JC + P"; then x„ + Pm = xm + Pm for w < n, in particular, x„ i P for any w e N. Again P + x„D = Z) and therefore x„D + P" = D. Hence there are elements u„ s D,v„ e P" such that JC„M„+VB = 1, or 1 + P" = x„u„ + v„ + P" = x„w„ + />" = (x„ + />")("« + P") = x„u„, where tin = un+P". Since the inverse element is unique, u„ + Pm = um + Pm for m < n, so (M„)„£N e J. Hence U(J) = {(xn)„GN\xi
*0>.
Let 6 =_(Z>„)„, and J/JP' m DIP'. The set {DIP', PIP',... ,P'-lIP',< 0 >} is the set of all ideals of DIP' by Lemma 4.2. Hence 7/JP' is isomorphic to P'~kIP' for some k. In other words, I = JP'"* = Jy' - *. This means that J is a principal ideal domain and every ideal of J has finite index. Let U be a non-zero J - submodule of V, U\ a J - pure envelope of U (i.e. U\IU is the J - periodic part of VIU). Since J - rank of V is r then rj(Ui) = rj(V) = q < r. We claim that U\IU is really finite. To show this, we note that VIU is a finitely generated J-module. Since J is a principal ideal domain, it is a noetherian ring. In particular, Ui/U is finitely generated. Now, we may observe that every finitely generated J- periodic module is finite, since every non-zero ideal of J has finite index. Now we want to prove that U\ f]A * < 0 >. Since \A/AP"\ = \VIVP"\, we can write that .4 + VP" = Kand.4 |~l VP" = AP", n e N. Therefore, VIU i = (A + Ui)/Ui +(VP" + Ui)/Uu and ((A + U\)IUX) n {{VPn + U\)IU\) = {AP" + Ui)/Ui. From VIUiP" = (VP" + U\)IU\ it follows that (V/Ui)/(V/Ui)P" = ({A + Ui)/Ui)) +{VP" + U\)IUX)I(VP" + U\)IUi) = ((A + Uy)IUX)l{{A + UX)IUy) fl {VP" + Ui)IUx)) = {{A + Ui)/Uiy{{AP" + Ul)/Ui) = {{A + Ux)IUx)l{{A + Ui)/Ui)P") s (4/(4 n Ui))/(A/(A f) Ui))P«. Since V = U\ © M, we have (VIU\)I(VIUX)P"
= M/MP", and MIMP" = DIP" © ... © DIP" . r-q
If we assume that A f l t / i = < 0 >, then (A/(A n Ui))/(A/(A n f/i))/"" = ^ / ^ / J " = DIP" © . © £>/P". r
This contradiction shows that^ fl U\ * < 0 >. Let t/ be a non-zero JG - submodule of V, U\ be a J - pure envelope of U in K Clearly U\ is a JG - submodule of K. As we have already shown, A(~\U\ is a non-zero DG - submodule of A, so the index \A : (A fl Ui)\ is finite. However, VIUi is J - torsion-free, so it follows that ,4 C\U\ = A.
Just Infinite Modules
66
Let a, be an element of A such that Bt\ = a\D + AP, 1 < / < r. Then V = a\J@ ... ®arJ. Since au... ,ar e Uu V=Uy. Thus \V: U\ = \U\ : U\ Consequently, every non-zero JG - submodule of Fhas finite index. Finally, the equation
is
finite.
VP" = lim{A/AP"+I | / e N} implies
r u ">»=. Hence J7 satisfies (JI 2). Thus Fis a just infinite JG - module. 6.7. Corollary Let D be a Dedekind Zo - domain, G a group, A a DG - module which is D - torsion-free, CQ(A) = < 1 >. If A is a just infinite DG - module, then there exists a field F > D and a simple FG - module B > A such that CQ(B) = < 1 >, and dim FB is finite. Proof Apply the above result. Let F be the field of fractions of the principal ideal domain J. On the other hand, B = V®j F. If E is a non-zero TO-submodule of B, then V C\ E is a non-zero JG-submodule of V. Hence the index | V : V C\ E\ is finite. In particular, the J - module B/(E C\ V) is periodic, so that and BIE is J periodic. From this, we must have E = B. In other words, B is a simple FG module. Since CG(V) = < 1 >, CG(B) = < 1 >. Finally, dimFB = rj{V), thus dimfB is finite. 6.8. Corollary Let D be a Dedekind Zo - domain, G a locally radical group, A a DG - module which is D - torsion-free, CG(A) = < 1 >. If A is a just infinite DG - module then G is abelian-by-finite. Proof Indeed, by Corollary 6.7 there are a field F > J and a simple FG- module B > V > A such that CG(B) = < 1 > and r = dimFB is finite. Under these conditions we can consider G as an irreducible subgroup of the linear group GLr(F). Since G is locally radical, then G is soluble (see, for example, [WB , Corollary 3.8]), therefore G is abelian-by-finite (see, for example, [WB , Lemma 3.5]). The statements 6.6 - 6.8 slightly generalize the main result of [KK]. The following result shows that in the study of just infinite modules it is
Just Infinite Modules over FC-Hypercenlral Groups
67
possible to apply the reduction to normal subgroups of finite index . 6.9. Proposition [FdeGK 3] Let R be a ring, G a group, A a just infinite RG module, CQ(A) = < 1 >. IfH is a normal subgroup of finite index ofG and X is a transversal to H in G, then A includes an RH - submodule B such that (i) AlBx is a just infinite RH - module for every x e X; (ii) A is isomorphic with an RH - submodule o / ® x e A . (A/Bx); (Hi) H is isomorphic to a subgroup ofXXsx(Hlx~lCx) Proof
where C =
CH(AIB).
Since A is a just infinite RG - module, it is noetherian. Put M = {Q\Q is an RH - submodule of A such that A/Q is infinite}.
Clearly M * 0. By [WJ 1, Theorem A] A is a noetherian RH - module. Hence the family M has a maximal element B. Obviously (~\xsX Bx is an RG - submodule of infinite index, therefore \^\xeX Bx = < 0 >. Assume that the RH - module AIB is not just infinite, and let AQ/B be the intersection of all non-zero RH - submodules of AIB. Then Ao * B, so A/Ao is finite. It follows that the RG- submodule A\ = |~|xeA- AQX has finite index in A. The RH - module AQIB is infinite simple, so AQX/BX is an infinite simple RH module for every x e X. Hence A\ + Bx = AQX for all x e X. By Remak's theorem, from the equation f*\xsX Bx =< 0 >we obtain the embedding A\ < © ^ 1 / ( ^ 1 l~l fix). Furthermore, A\I{A\ C\Bx) £ (A\ + Bx)IBx = Aox/Bx is a simple RH module, in particular, A i is a semisimple RH - module. Since every RH - module A\I{A\ fl Bx) is infinite, then A i includes an infinite simple RH - submodule E. If S is a non-zero RG- submodule of A, then AIS is finite, so EI(E fl S) is finite too. Thus E = E f| S, since E fl S is an RH - submodule of E, that is E < S. Hence the intersection of all non-zero RG - submodules of A includes E. However, this intersection is zero. This contradiction shows that Ao = B. In other words, AIB is a just infinite RH - module. It follows that AlBx is a just infinite RH - module for each x e X. Once more, from Remak's theorem we obtain the embedding A
, and a(c - 1) = 0, so c e CG{A) =< 1 >. Hence the subgroup C cannot contain the q>- elements. Furthermore, the equation |~)neN^2" = < 0 > shows that HneN CG(AIAQ") < CG(A) = < 1 > . Together with Corollary 5.23 this implies that G is residually finite. Suppose that charD = 0; then charF = 0. Since G < GL„(F), all Sylow q subgroups of G are Chernikov for any prime q (see, for example, [WB, Theorem 9.1]). Moreover, they are finite (G is residually finite). We have already proved that Cu{AIAQ) does not contain the ql- elements. Since UICu(AIAQ) is finite, this means that the periodic part T of the abelian subgroup U is finite. By the residual finitenes of G it follows that U includes a G - invariant torsion-free subgroup of finite index (G is residually finite). Finally, let char D = p > 0. Similarly we can obtain that U includes a G invariant torsion-free subgroup of finite index. Suppose that Op(G) * < 1 >. Since G is an FC - hypercentral group, Corollary 3.4 yields that Op(G) fl FC(G) * < 1 >. Hence Op(G) includes a non-identity finite abelian G invariant subgroup E. Put W= CG(E); then W is a normal subgroup of finite
Just Infinite Modules oxer FC-Hypercentral Groups
73
index. Proposition 6.9 shows that A includes a DW - submodule B\ such that^4/5i is a just infinite DW- module. Since AIB\ is an elementary abelian p - group, it is easy to see that CA/B,(.E) * < 0 >. Since E is abelian, E < C,{W). Corollary 6.5 yields that in this case E < CG(AIB\). Let S be a transversal to W in G. Then E = x~lEx < x~xCw{AIB{)x
=
Cw{AIB\x)
for each x e S. By Proposition 6.9 A < @xeS(A/Bix). From this embedding it follows that E < CG(A) = < 1 >, a contradiction. Thus Op(G) = < 1 >. 6.16. Corollary Let D be a Dedekind ZQ - domain, G an FC - group, A a just infinite DG - module which is D - torsion-free. IfCdA) = < 1 >, then G/£(G) is finite. Moreover, C,{G) includes a torsion-free subgroup of finite index. If charD = p > 0, then Op{G) = < 1 >. Proof By Theorem 6.15 G includes a normal abelian torsion-free subgroup of finite index. Since G is an FC - group, it is central-by-finite (see [TM, Lemma (7.5)]). The other assertions are either trivial or are contained in Theorem 6.15. 6.17. Corollary Let D be a Dedekind ZQ - domain, G a locally radical group, A a just infinite DG - module which is D - torsion-free. IfCdA) = < 1 >, then G includes a normal abelian torsion-free subgroup of finite index. Moreover, if charD = p > 0, then Op{G) = < 1 >. In fact, G is abelian-by-finite by Corollary 6.8. In particular, G is FC hypercentral. The next results shows that in some special situations it is possible to realize the reduction to the torsion-free case. 6.18. Proposition LetD be aDedekindZo - domain, G a group, x an element of infinite order of the center £(G), A ajust infinite DG - module, CG(A) =< 1 >. If A is D - periodic, then AnnD(A) = P e Spec{D). IfF = DIP, then A is F < x > torsion-free. Proof Corollary 6.4 yields that Anno(A) = P e Spec{D). Hence we can consider A as FG - module where F = DIP is a finite field. Put J = F < x > and think of A as a JG - module. Suppose that A contains an element 0 ± a s A such that / = Annj(a) * < 0 >. Since every non-zero ideal of J has finite index in J, aJ = JII is finite. It follows that x' e CQ{A) for some t e N, that is x' - 1 el. Since 0 * a e AnnA{I) < AnnA(x' - 1), this contradicts Corollary 6.5. This contradiction shows that A is J- torsion-free.
Just Infinite Modules
74
Summing up these results, we deduce some consequences, which we simply state. 6.19.Corollary Let D be a Dedekind Zo - domain, G an FC -hypercentral group, the center of which contains elements of infinite order, A a just infinite DG - module, CG(A) = < 1 >. If A is a D -periodic module, then Anno(A) = P e Specify) and G includes a normal abelian torsion-free subgroup of finite index. Moreover, Op{G) = < 1 > where p = chariDIP). 6.20.Corollary Let D be a Dedekind Zo - domain, G a locally radical group, the center of which contains elements of infinite order, A a just infinite DG module, CQ(A) = < 1 >. If A is aD -periodic module, then Anno{A) = P e Spec(D) and G includes a normal abelian torsion-free subgroup of finite index. Moreover, Op(G) = < 1 > where p = char{DIP). The setting of these last results is that £(G) contains infinite cyclic subgroups and, in particular, f(G) * 1. We are finishing this chapter showing that this condition cannot be removed, as the next example shows. 6.21 Example Let p be a prime, C = Q C =< c„ \ {c„+\)p = c„, n e N >, < x > an infinite cyclic group, G = C X< x > wherex~ l c„x = c„+\,n e N. Let q be a prime such that q * p. Consider the group ring ¥qG. Suppose that / is a non-zero G - invariant ideal of the ring ¥qC. Since ¥qC = | J „ g N F ? < c„ >, there exists a number k such that lC\¥q < c* > = I\ * < 0 > . The factor - ring F ? < Ck > II\ is finite, so there is a number t e N such that c\ - 1 6 I\. In other words, a subgroup C contains an element c such that c - 1 e I\. If C\ =< c >G , then the index \C : C\\ is finite. Consider the ideal h of the ring ¥qC, generated by the elements b-\ where b e Ci. From the equation vy - 1 = (v - l)(y - 1) + (v - 1) + (y - 1) it follows that I2 < I. Since CIC\ is finite, ¥qC/I2 is finite and ¥qCII is finite too. Let R be a right ideal of ¥qG, generated by elements x" - 1, n e Z, and L be a right ideal of ¥qG with the properties: L > R and L * R.lfu e ¥qG, then u = a\C\xh + ... + a„c„x'", a, e ¥q, ci G C, tt e Z, 1 < i < n. It is clear that u = a\C\{xh
- 1) + ... + a„c„(x'n - 1) +a\C\ + ... + a„c„,
and therefore ¥qG = R + ¥qC, so that L = R + (L f] ¥qC). Since L=t R,h *< 0>. If 6 e 7 3 , then
Put h = L D F ? C .
Just Infinite Modules oxer FC-Hypercentral Groups x-mbxm
=
_(xm _ l)( JC -'")fc c '« +
75
fo^
that is i " * f e " e l . Since J e F,C, b = P\d\ +... + psds, where pj e F,, of, G C, 1 <j< s. Then jc^fec* = P\(x-mdixm) + ... + ps{x-mdsxm)
e F,C.
Hence x~mbxm e i f l F ? C = /3, which allow us to establish that h is a G invariant ideal of ¥qC. As we have already proved, FqC/h is finite, which gives that the index \VqG : L\ is finite too. Consequently, A = FqG/R is a just infinite F,G - module. It is easy to check that R f| F ? C = < 0 >, which assures that if 1 * c e C, then c - 1 £ /?. Suppose that c e CG(^I). Let g e G, then (g + R)c = gc + R = g + R, that is g(c - 1) e /?. Therefore g_1g(c - 1) e R, a contradiction. Thus CGOO n C = < 1 >. Since CG{A) is normal in G, it follows that Ca(A) = < 1 >. However, G is not abelian-by-finite.
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Chapter 7 Just Infinite Modules over Groups of Finite O-Rank
In the previous chapter we began to consider just infinite modules over an FC hypercentral group G. As we have proved, in this case G is abelian-by-finite. However, we obtained virtually no information about the structure of these modules. Now we will consider the case in which G is a FC- hypercentral group withfinite0-rank. In this case we may obtain some additional specific information about both the group G and the structure of just infinite modules over G. We shall need some assumptions about the underlying ring of coefficients. Actually, a Dedekind Z\-domain is a Dedekind Zo-domain in which Spec(D) is an infinite set. 7.1. Lemma Let D be a DedekindZi-domain, G a locally (polycyclic-by-finite) group of finite 0-rank, A a DG - module, which is D - torsion-free and A = MDG, for some finite subset M. Suppose that H is a finitely generated subgroup ofG of the same 0-rank, ro(H) = ro(G), and let B = MDH. If the factors AIAP are finite for P e 7r c Spec(D) and the set n is infinite, then ro{B) is finite. Proof Fix Pen: and put R = CH{AIAP). Since His polycyclic-by-finite, then R is finitely generated. Suppose that/? = < x\,... ,x, > andM= {a\,... ,a„}. There are elements uy € AP such that atXj = a, + Ujj, 1 < i < n, 1
<j2 e D be generators of P: P = y\D+yjD. Hence uy = Vyy\ +Wyy2, where v,j,Wij e A. Choose a finitely generated subgroup R\> H satisfying the condition v,y,wy e MDRi, whenever 1 < i < n,\ < j < t. Put E = MDR, E\ = MDR\ and
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£i = E\IE\P. Then we have a, + E\P = a, = (a,)*/, so that atDR = atD, 1 < i < n. This means that E = (E + E,P)IEyP = ZasM^R
= "Z^am
= £lsSia,Z>.
Since DIDP is finite, so is E. Since the index \R\ : R\ is finite, Lemma 1.14 implies that E\ is finite too. Now |/?i : H\ is finite, so that E\ is a finitely generated Z)//-submodule and [WJ 1, Theorem A] actually yields that E\ is a noetherian ZW-module. Let Y/BP be the P-component of E\IBP. By Lemma 1.9, AnnD(Y/BP) * < 0 > Clearly, £ < y. Take an integer / > 0 such that (JIBP)P' = < 0 > and call Y\IBP = (EilBP)P'. There is a D - submodule LIBP such that Ei/BP = (Y/BP) © (LIBP) [KI], and so (Y/BP) D (Y\IBP) = 0. We have already proved that E\IE\P is finite. Since E\ is Z)-torsion-free, Corollary 5.23 implies that Ei/EiPm is finite for any m e N (the finiteness of DIP implies that DlPm is finite, m e N). In particular, (E\IBP)I(Y\IBP) is finite, so that YIBP and BIBP are finite. Corollary 1.8 then gives that B e A(D,K\) for some finite subset K\ £ Spec(D). By Lemma 1.6 r D (B) = dimD/P(BIBP) for P i Ki. But n is infinite, so that n\it\ is infinite too. Since BIBP is finite for P e ;rWi, we may conclude that ro(B) is finite. Let D be a Dedekind domain. If A is a D-module of finite D-rank and M is a maximal Z)-free subset of yL then^/MD is a D-periodic. Put^o = MD and the set SPD(A) to be the set of all P e Spec(D), for which the P-component of AlAo is not bounded (that is its annihilator in D is zero). If B is another free Z)-submodule of A and ro(A) = rp(B), then AQ/(AO 0 B) and B/(Ao f) B) are finitely generated D-periodic modules, so AnnD(Aol(Ao P\B)) and AnnD(B/(A0 f]B) are non-zero, which shows that SPD(A) is independent of the choice of the free submodule Ao. In the sequel, we write Sp(A) instead of Spz(A). A module A is called a minimax module if it possesses a finite series of submodules with either artinian or noetherian factors. Let R be a noetherian ring, A a minimax i?-module. Take a finite series < 0 > = A0 . It follows that A e A(D,7t), where n = YlD(AIC) is a finite set of Spec(D). Hence, if A is a minimax £>-module, then SPD{A) is finite and, with the above notation, SPD(A) C HD(A/C). Furthermore, if A is torsion-free, we may choose a D-submodule C such that SPD(A) = TID(A/C). If D is a field, we simply note that a minimax D-module has finite dimension (as a £>-space). .4M abelian group A is called minimax, if the Z - module A is minimax. 7.2. Lemma Let D be a Dedekind domain, G a polycyclic-by-finite group and A a finitely generated DG-module of finite D-rank. If A is D-torsion-free, then A is a minimax D-module. Proof By Corollary 1.8, A e A(D,n), for some finite subset n of Spec(D). In other words, there exists a projective submodule C < A such that AIC is a n periodic module. In particular, ro(A) = ro(C). Take P e Y1D(A/C) and let YIC be the P-component of AIC, Y\IC = Q^^F/C). Since Y\ has finite Z) - rank, then dimDip(J\IY\P) is finite and, in particular, dimn/p(Yi/C) is finite. Then, YIC is an artinian £>-module by Lemma 5.6. Since IID(A/C) is finite, then AIC is artinian too. Thus A is D - minimax. 7.3. Corollary Let D be a Dedekind domain, G a polycyclic-by-finite group, A a finitely generated DG-module, M a finite subset A, H a subgroup ofG. IfroiA) is finite, then the submodule B = MDH is D - minimax andSpoifi) c: Spo{A). 1.4. Lemma Let D be a Dedekind domain, G a group, H,K normal subgroups ofG, A a finitely generated DG-module and Ma finite subset of A such that A = MDG. lfrD(MDH) andrD{MDK) are finite, then rD{MDHK) is finite. Proof Since ro{MDH) is finite then M includes a finite subset X such that (MDH)/(^laeM ^axD) is D - periodic. Similarly, K includes a finite subset Y a such that (MDK)/(£ eY yD) is £> - peniodic. The set FZis finite, therefore it is sufficient to show that (MDHK)/(Y> ^, azD) is D - periodic. Let a e M, h e H, g e K. There is an element u e D such that
{ah)u e Y,t
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and therefore (ahg)u G X) Let b G Mand consider the element b(xg~lx~x). Since AT is a normal subgroup of G, xg~xx~x e K. Thus there exists an element v e D such that bixgx-^v e E ^ j ^ a ^ Then
It follows that there is an element w e D such that {ahg)mv e X ^ ^ a z D .
D
7.5. Lemma Le? D be a Dedekind Z\ -domain, G an abelian group of finite 0-rank and A a just infinite DG - module. If A is D-torsion-free and charD = 0, then ro{A) is finite. Proof Let //be a finitely generated subgroup of G such that ro(H) = ro(G) and let M be a finite subset such that A = MDG. Put E = MDH. By Theorem 1.15, there is a subset n £ Spec(D) such that Spec(D)\n is finite and ^4 * AP for each Pen. Thus Lemma 7.1 and Lemma 7.2 yield that ro{E) is finite and £ is a minimax Z)-module; in particular, SPD(E) is finite. Since n is infinite, n\SpD(E) * 0. Consider J° e n:\SpD{E). Let/» = char(DIP) and Q/Hbe a Sylow p'-subgroup of G///. We claim that rD(MDQ) is finite. Since ,4A4P is finite, GICG{AIAP) is finite too. Put //i = CH(A/AP) so that /////i is finite. By Corollary 7.3, MDHX is a minimax /^-module and SPD(MDH\) C Spo(E). In particular, P g SpD(MDHi) and we may assume that / / = / / , , i. e.// < CG{AIAP). Let /. As a consequence, we deduce that the characteristic of the finite fields DIP, Pen, cannot be a constant. Otherwise, if char(D/P) = p for each P e n, A could be embedded in the Cartesian product Ylp^AlAP, so pA = 0, while charD = 0. Therefore, there exist Pi,P2 £ ff such that char(D/P\) * char{DIPi). Further, we may assume that Pi,P2 £ Spn(E). For / = 1,2, let Qj/Hbe a Sylowp,' - subgroup of GIH. Then G = 0 i 2 2 and, since we have already proved that roiMDQi) are finite, then A"D(^) is finite by Lemma 7.4, as required. The above lemmas are the reformulated versions of some results of [ZKT]. 7.6. Theorem Let D be a Dedekind Zi-domain of characteristic 0, G an FC-hypercentral group offinite0-rankandA a just infinite DG-module which is D - torsion-free, CG(A) = < 1 >. Then ro{A) is finite and G includes a torsion-free abelian normal subgroup of finite index. Moreover, ifK is the field of
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fractions ofD andn = ro(A), then G is isomorphic to an irreducible subgroup of GL„(K). Proof By Theorem 6.15, G includes a torsion-free abelian normal subgroup H of finite index and, by Proposition 6.9, there exists a ZW-submodule B such that AIB is a just infinite DH - module. If X is a transversal to H in G, it follows that AlBx is just infinite for every x e X, and A embeds in @xeXA/Bx. By Lemma 7.5, each summand of the above direct sum has finite Z)-rank. Since X is finite, ro(A) = n is finite. If E = A ®o K, we note that n = dim^E and CG(E) = CG(A) = < 1 >, so that G can be considered as a subgroup of GL„(K). This subgroup is irreducible, because E is an irreducible ATG-module. 7.7. Corollary Let G an FC-hypercentral group of finite 0-rank and A a torsion-free just infinite ZG-module with CG(A) = < 1 >. Then (1) G is a finitely generated abelian-by-finite group; (2) the additive group of A is a minimax. Proof By Theorem 7.6 G is an abelian-by-finite irreducible linear group over Q and [CV 3] implies that G must include a finitely generated free abelian subgroup of finite index, then showing (1). (2) follows from Lemma 7.2. 7.8. Corollary [KK] Let G a locally radical group of finite 0-rank and A a torsion-free just infinite ZG-module with CG(A) = < 1 >. Then (\)Gis a finitely generated abelian-by-finite group; (2) the additive group of A is a minimax. Proof
It suffices to apply Corollary 6.8 and the above result.
Let G be an abelian-by-finite group of finite 0-rank and let H be an Abelian subgroup of G of finite index. In the next results, we shall put Sp(G) = Sp(H). 7.9. Theorem Let D be a Dedekind Z\-domain with charD = p > 0, G an abelian-by-finite group of finite 0-rank, A a just infinite DG-module, CG(A) = < 1 >. If A is D - torsion-free andp £ Sp(G), then ro(A) is finite. Moreover, G is isomorphic to an irreducible subgroup of GL„(K) where K is the field of fractions for D, n = ro(A). Proof As in the proof of Theorem 7.6, it suffices to assume that G is abelian. Let / / b e a finitely generated subgroup of G such that ro(G) = ra(H) and let M b e a finite set of generators of A : A = MDG. Since p H such that QIH is ap'-group and \G : H\ is finite. A verbatim repetition of the arguments given in Lemma 7.5 and in Theorem 7.6 proves the finiteness of ro(A) and the
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required embedding of G. Given a group G; if A is a Z)-periodic just infinite DG-module, then, by Corollary 6.4 Anno(A) = P e Spec(D). If there exists an element x e £(G) having infinite order, then by Proposition 6.18 A is F < x >-torsion-free, where F = DIP is a finite field. Thus, we come back to the torsion-free case where D = F < x > is the group-ring of an infinite cyclic group < x > over a finite field F. In this case, the following lemma is almost obvious. 7.10. Lemma Let D = F < x >, where F is a finite field, \F] = q, < x > is infinite. Given k e N,we consider the map , ifD = F < t >, then A can be viewed as a DG-module defining at = azfor each a e A, and, with this meaning, there exists a finite subset n £ Spec(D) such that A e A(D,n). Proof If / = AnnfG(A), then A = FGII. Since / is a prime ideal, the factor-ring R = FGII is an integral domain. Then the mapping
. Put B0 = < 0 >,BX = AH"-l,...,B^i
= AH,B„ = A.
Clearly the series Bo < Bi < ... < £„ = A is RG- central. 7.13.Theorem Let F be a finitely generatedfield,A a finite dimensional vector space over F.IfG is a soluble automorphisms group ofA, then G has a series of normal subgroups < 1 > < H < E < G, where (1) G/E is finite; (2) EIH is a countable free abelian group; (3) H is a nilpotent subgroup; (4) if char F = 0, then H is torsion - free; if charF = p > 0, then H is a boundedp - subgroup; (5) A is an FH - nilpotent module. Proof We can consider G as a subgroup of GL„(F) where n = dimpA. By Maltsev's theorem (see, for example, [WB, Theorem 3.6]) there are an element g e GL„(F) and a normal subgroup £o of G such that the index \G : Go\ is finite and g'l(Eo)g< T„(F) where F is an algebraic closure of the field F. Let g = ||ay||i = Z 0 < Z i < ... < Z„ =Ai is F i / / - central. In other words, Ai is F i / / - niipotent, so that A is an FH niipotent module. The group UT„{Fi) has a central series UT„(Fi) = UT$\Fi)
> UTf\Fi)
> ... > UTin)(Fi)=
,
where UTim)(Fi)
= < t,j(a) | a e Fuj - i > m >, t0{a) =
E+aE0,
(see, for example,[KMM, 16.1.2]). Moreover, U&\Fi)IUlt+l){Fi)
=F\ x ... x F\
(here each factor F\ is the additive group of the field Fi, [KMM, Ch. 4]). In particular, Hi is niipotent. Moreover, if charF = 0, then F\ is torsion - free, so that Hi is torsion - free. If char F = p > 0 then F\ is an elementary abelian p group, so that Hi is a bounded p - group. Since H and Hi are conjugate, the same is valid for subgroup H. Finally, T„(Fi)IUT„{Fi)
= U(Fi) x ... x U(Fi) .
Theorem 4.10.1 from [KG] yields that U(Fi) is a direct product of finite cyclic subgroup and countable free abelian subgroup. It follows that Eo includes a G invariant subgroup E such that \G : E\ is finite and EIH is a countable free abelian group.
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7.14. Corollary Let F be a finitely generated field, G a soluble group, A a simple FG - module, Ca(A) = < 1 >. IfdimpA is finite then G is a finite extension of a countable free abelian subgroup. Proof
By Theorem 7.13 G has a series of normal subgroups . 7.15. Corollary Let R be a finitely generated integral domain, A an R - module of finite R - rank which is R - torsion - free. If G is a soluble automorphisms group of an R - module A, then G has a series of normal subgroups RF. We can uniquely extend the action of G on A to an action of G on B. Since rR(A) is finite, dimpB - r^iA) is finite too. Now we can apply Theorem 7.13. 7.16. Corollary [ZKT] Let F be afinitefield,< t > an infinite cyclic group, J = F < t >, A a J - module of finite J - rank which is J - torsion -free. If G is a soluble automorphisms group of the J - module A, then G has a series of normal subgroups Fitt(G(,) such that \Gj : Go\ is finite. Therefore Go = (Fitt(Ge))L where L = Go fl L\ is a nilpotent subgroup. From now on, we are considering just infinite FG - modules, where G is a primitive polycyclic group and F is a finite field. In this case it is worth mentioning that the condition (JI 2) from the definition of the just infinite module is automatically satisfied: indeed by [PD 1, Theorem 12.3.7] a simple FG module is finite. Let G be a polycyclic-by-finite group. A subgroup P is called a plinth of G if the following conditions are sufficed: (PL X)P is a non-identity torsion-free abelian subgroup ofG; (PL 2) if H= NG(P), then the index \G : H\ is finite and HICH(P) is abelian-by-finite; (PL 3) if S is a subgroup of H having finite index in H, then P ® zQ is a simple QS - module. (See, for example, [PD 1, Chapter 12, Section 3]). Every polycyclic-by-finite group includes a plinth (see, for example, [PD 1, Lemma 12.1.4]). Let G be a primitive polycyclic grou. Choose a non-identity free abelian normal subgroup P of G of the smallest possible rank. Replacing P by a larger subgroup if necessary, we may assume that P is contained in none abelian normal subgroup of G as a proper subgroup of finite index. Let R < P and suppose that \G : NQ(P)\ is finite. Since G is primitive, PJCorea(R) is finite, and therefore \P : Corea(R)\ is finite too. Since P z Q is a simple QG - module then
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G/CG(P) is abelian-by-finite (see, for example, [WB, Lemma 3.5]). This means that P is a plinth of G. Consequently, a plinth of a primitive polycyclic group is a normal subgroup. Since the case of the group G with f (G) * < 1 > has been previously considered, we can consider only the case of the group G with £(G) = < 1 >. 8.2. Lemma [RW] Let F be a finite field, G a primitive polycyclic group with £(G) = < 1 >, P a plinth ofG, P is normal in G,R = FP. If A is a just infinite FG - module with Ca(A) = < 1 >, then A as an R - module is torsion-free and has finite R - rank. Proof Suppose that A has an R - periodic element a * 0. Then Annn(a) * < 0 >. We have a e AnnA(Annn(a)). Let •A4 = {Q | Q is a non-zero ideal of if such that Ann A{Q) * < 0 >}. Then M =£ 0. The group ring R is noetherian (see, for example,[PD 1, Theorem 10.2.7]). It follows that M has a maximal element /. Put B = AnnAif). Let I\,h be ideals of R such that I < h, I < h and l\h = I- From B(I\h) =< 0 > we obtain that either AnriA(I\) * < 0 > or AnnA(Ji) * < 0 >. By the selection of/, either is / = h or / = h- In other words, / is a prime ideal of/?. If x e G, then Bx = AnnA(Ix). In fact, let B\ = AnnA(Ix). If b e £,>> e 7, then
(&x)(x~1.y*) 1
=
(fyO* = o>
x x
that is fix < Tii. Hence (fii)x" < AnnA((I ) ~') = B, so fix = fii. If x e NG(I), then fix = AnnA(Ix) = AnnA{I) = B, thus we can consider B as an F(NG(I)) - submodule. Let T be a transversal to MsCO in G. Then B(FG) = ^ gT-^x- Suppose that this sum is not a direct sum. Then there are elementsx\,... ,X£, Xk+\ £ Tsuchthat Bxi+i n (fixi + ... + Bxk) * < 0 >. Put> .
Since ArmA{I+ (/>" n ... n / w ) = ^Iw^CO 0 (AnnAUy')
+ ... + AnnA(P"))
=Bf](By
from the choose of / we obtain the inclusion H I - S K * ^ ' - ^' anc * t n e r e f ° r e Iy'...In < I. Since / is a prime ideal, P" < I for some i. However, in this case
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I < P~> . Taking in account that Ann A{F ) = ByJ1 =t=< 0 >, by the choice o f / w e can conclude, that / = pi . This implies that yt e NG{I), a contradiction. Consequently, B(FG) = ®xsTBx. Put H = NG(I). Assume that B includes a proper non-zero FH - submodule C. Then C(FG) = (BxeTCx. Since A is a just infinite FG - module, AIC(FG) is finite. This means that the set T is finite. In other words, the index \G : H\ is finite. The subgroup P is a plinth in H too. By Bergman's Theorem (see, for example, [PD 1, Corollary 9.3.9]) the factor-ring R/I is finite. It follows that there exists a number m eN such that 1 -ym e I for each y e P, that is Pm < CG(B). Since Pm is normal in G, pm
= x-\pmx
< X-1CG(B)X = CG(Bx)
for every x e T. It follows that /"" < CG{B{FG)). Since the A/B(FG) is finite, there is a number mi e N such that /""' < CG(A/B(FG)). Since the additive group of ^ is an elementary abelian p - group, there is a number mj e N such that /"" 2 < C G 0 4 ) =< 1 >, and we obtain a contradiction (G is torsion-free). If we assume that B is a simple FH- module, then B is finite (see, for example, [PD 1, Theorem 12.3.7]). Thus there is a number m 3 e N such that Pm e CG{B), and again we come to a contradiction. Consequently, A is R - torsion-free. By Theorem C from [RJ 1] there are a free R - submodule E of A and a non-zero ideal A of R such that every element oiAlE is annihilated by some product A*' ... A*" for suitable x\,... ,x„ e G. Since P is a plinth of G, there is a maximal ideal L of R which includes no conjugate of A by Theorem E from [RJ 1]. Corollary CI from [RJ 1] yields that^ = E + AL and EL = EC\AL. In particular, AIAL and EIEL are isomorphic as R - modules. Since L is maximal in R, the factor-ring RIL is finite, so P/Cp(PJL) is finite. Hence for some / G N the ideal L generated by all elements xl - l,x e P, lies in L. It is clear that L is a non-zero G - invariant ideal of R. Therefore AL is a non-zero FG - submodule of A (A is R - torsion-free). It follows that AIAL is finite, hence AIAL is finite too. EIEL is finite sinceAIAL =R EIEL. The R - module E is free, that is E = ® a s r £« where Ea = R for every a e T. Then jE/isZ, = ® a e r EJEaL where EJEaL = RIL. It follows that T is finite. In other words, the R - module E has finite R - rank. Since P is normal in G, A*1... A*" < R. that is A*1... A*" < AnnR(A/E). Thus ,4/JE is /? - periodic, and therefore the R - module A has finite R - rank. 8.3. Lemma [RW] Let F be a finite field, G a primitive polycyclic group with C,(G) =< 1 >, P a plinth ofG and also P is normal in G. Let A be a just infinite FG - module with CG(A) = < 1 >. Then P is a maximal normal abelian subgroup ofG. Proof
Put R = FP. Suppose that there is a normal abelian subgroup P\> P
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suchthatPi * Pandlet^i = FP\. Let A = AnnRl(A). By Lemma 8.2 A * Ru It is clear that A is G - invariant. Let x e R\\R. Then x induces an R - linear mapping in A. If a e A, then aR<x>=R<x>
/AnriR<x>(a).
Since the R - module^ has finite R - rank, AnriR<x>{a) * < 0 >. If a\,... ,ar are the maximal R - independent subset of A, then P| 1. Suppose that there are G - invariant ideals I\ and h of R\ such that I\h < A, but I\ $ A, h $ A. If y e I2, then ^(/i>0 = < 0 > . Thus the mapping a + AI\ —• ay,a e A, is an R\ - homomorphism of AIA(I\) with the image Ay. Since I\ S A and I\ is G - invariant, ^(/i) is a non-zero FG - submodule of A. It follows that AIA(I\) is finite, so Ay is finite too. The ring R\ is noetherian, so there are elements y\,... ,y, e h such that h = Riy\ +...+R\yt. Therefore A(h) = Ay\ +...+Ayt is a non-zero finite FG - submodule of A. This contradiction shows that such ideals I\ and h do not exist. By Lemma 5 of [RJ 2] A = Ai fl ... fl Am where Ai,... , Am is a complete G - orbit of the prime ideal of Ru Let K = (1 + Aj) n Pi. Then { F | g e G } = {(1 + A,) n ^1 | 1 < i < m}. Since G is a primitive polycyclic group and the index \G : NQ{K)\ is finite, KICoreG(K) is finite. If u e CoredK), then M - 1 e Ai D - fl Am = A, so /l(w - 1) = < 0 >. Since CG(A) = < 1 >, Corea(K) = < 1 > . Hence K = < 1 >, since G is torsion-free. Consequently, Ai is a prime ideal of Ri such that the index \G : NG(AI)\ is finite and (1 + Ai) n Pi = < 1 >. By Theorem D from [RJ 2] Ai = (Ffl Ai)Ri, so that Ai = < 0 >. In particular, A = < 0 >, a contradiction. 8.4. Lemma [RW] Let F be afinitefield, G a primitive polycyclic group with C(G) =< 1 >, P a plinth ofG and also P is normal in G. Let A be a just infinite FG - module with CG(A) = < 1 >. Then G splits over P : G = P X T where T is a free abelian subgroup, CT(P) = < 1 > and P ®z Q is a simple QT- module. Proof Put R = FP and let K be a field of fractions of R, B = A®RK. By Lemma 8.2 dimKB is finite. Put H = CG(P)- Then P possesses a finite series < 0 >= Bo < Bi < ...
. Lemma 8.3 yields that CG{P) = H=P. Since P < Fitt(G), C = £(Fitt(G)) < CG(P), thus C < P. It follows that PIC is finite, since P is a plinth of G. Since Fitt(G) is torsion-free, Fitt(G)IC is torsion-free too (see, for example, [RD 19, 5.2.19]). This means that P = C, therefore P = F/«(G) because P = CG(P). Let T be a nilpotent subgroup with the property G = PT. Suppose that P f)T * < 1 >. Since POT is normal in 7, L = f (7) n P * < 1 >. Since G = P77, the subgroup L is normal in G. Then P/Z, is finite. It follows that [P, 7] is finite, thus [P, T\ =< 1 > and 7" < C G (P) = P, a contradiction. It shows that P f] T = < 1 >. Since P is a normal abelian subgroup of the smallest possible rank, P ®z Q is a simple Q 7 - module. In this case T/CT(P) is abelian-by-finite (see, for example,[WB, Lemma 3.5]). Hence T is abelian-by-finite. It follows that T is abelian, because 7 is torsion-free and nilpotent. From Lemmas 8.2 - 8.4 we obtain 8.5. Theorem [RW] Let F be a finite field, G a primitive polycyclic group with C,(G) =< 1 >, A a just infinite FG - module with CG(A) = < 1 >. Then (1) P = Fitt(G) is an abelian subgroup; (2) P is a plinth of G; (3) G = P X T, where T is abelian, CG(P) = P, P ®z Q is a simple QT module; (4) the FP - module A is torsion-free and has finite R - rank. The aforesaid paper also describes a method for constructing just infinite modules, which we recall now. Let G = P X T be a primitive polycyclic group, P = Fitt(G) a plinth of G, CG(P) = P, P and T finitely generated torsion-free abelian subgroups, P ®% Q a simple Q 7 - module. Put R = FP and let .K be the field of fractions of R. If r e N, then let AT(r) be a /^ - vector space formed by all r - tuples over K and let R^ the corresponding R - submodule of all r - tuples over R. The action of elements of T on P by conjugation may be extended in the obvious way to the actions on R and K. Then Tacts on P w and K^ through its action on components, and these action can be extended to GLr(K) through its entries.
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Now choose a derivation from Tto GLr{K), that is a function 8 : T — GLr(K) with the following condition: (txh)s
= (r?)"(f2)*
forallfi,; thus both \B : B n L\ and \A : L\ are finite. Conversely, assume that A is just infinite. We regard A as embedded in V - A ®A K which is an r - dimensional K - vector space. The action of T on A extends naturally to Fif we define (a ®f)t = (at) ®f where a eA,fe
K,t e T.
It is easy to check that according to this definition V becomes an FG - module. Choose a basis {e\,...,er} of V consisting from elements of A. For each t e T we have
where fyit) e K. Put
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ts = MOWThen ts e GLr(K), because the elements e\t,... ,ert are linearly independent. Let v e V,v = YjiF(P )) is a non-zero FG-submodule for each m e N. It follows that A/A(coF(Pm)) is finite. If Vo is a non-zero %T- submodule of V, then A fl Vo is a non-zero FG - submodule of^4. Hence A/(A f\ Vo) is finite and therefore Vo = K. Conversely, assume that conditions (1) and (2) are satisfied. Since A is finitely generated over FG and the ring FG is noetherian (see, for example, [PD 1, Theorem 10.2.7]), A is a noetherian FG - module. Let M = {U | [/ is an FG - submodule such that AIU is infinite }. It is clear that M * 0. Then ,M has a maximal element B. Let 5 = {U | [/ is an FG - submodule such that U > B and U * B}, B0 = f\ S. Then AIU is finite for every U e S. If B = Bo, then A/B is a just infinite module. If Bo * B, then A/Bo is finite. Hence Bo/5 is an infinite simple FG - module. However, for every polycyclic group G and every finite field F each simple FGmodule is finite [RJ 1, Theorem A]. This contradiction shows that A/B is a just infinite FG - module. Suppose that B * < 0 >. If A/B is not R -periodic, then B\IB be the R - periodic part of A/B. Clearly, B\ is an FG - submodule of A. Since B\ * B, rn(B\) < rR(A) and, therefore, B\RK is a proper non-zero T - invariant subspace of V. This contradiction proves that A/B is an R - periodic module. Let H = CG(A/B). Lemma 8.2 yields that H * < 1 >. Thus //fl P * < 1 > and P/(P fl H) is finite. It follows that Pm < H for some m e N, so that A(aF(Pm)) < B. Then A/B is finite by (2). This contradiction shows that B = < 0 >, i.e. A is a just infinite FG - module. If A is finitely generated by n elements as an R - module , then Ap* is a module over the finite ring F(PIPm)\ clearly Ap« has finite order at most ql where / = n\P : Pm\,q = \F\. It now easily follows that if every finitely generated R submodule of A is contained in a submodule generated by n elements, then Ap* is finite. A derivation 8 : T — +AI)/AI) = (A/AT)/(APj"'/AT) = A/AP,"' is finitely generated, giving that A/AI is finitely generated too. The converse statement is obvious. 9.2. Lemma Let D be a Dedekind domain and let A be a D-module. Then A is D-co-(layer-finite) if and only ifA/Ax isfinitelygenerated, for every 0 * x e D. Proof. The direct assertion is trivial, using Lemma 9.1. For the converse .it suffices to recall that an ideal of a Dedekind ring can be generated by at most two elements (see, for example, [KG, Corollary 3.3.14]). 9.3. Lemma Let D be a Dedekind domain, A a D - module, B a D - submodule ofA. (1) If A is co-(layer-finite), then so isA/B. (2) IfB andA/B are co-(layer-finite), then and A is also co-idayer-finite). Proof (1) is obvious. (2) Let / b e a non-zero ideal of D. Then (A/B)/((AI + B)IB) is finitely generated. Since (AI + B)IAI s B/(B n AT) and
BI
where Ap = Bp © Cp is the P-component ofA, Bp is finitely generated and Cp is divisible. 9.7. Lemma Let D be a Dedekind domain and let A be a torsion-free D-module having an ascending chain ofD-pure submodules = A0, where Fitt{G) is the Fitting subgroup of G, that is the subgroup generated by all normal nilpotent subgroups. Naturally, the first step here is investigation of a structure of the Fitting subgroup of just non-A" -groups. In fact, for groups from all indicated classes it is possible to prove that its Fitting subgroups are abelian. This allows us to consider them as modules over the group GIFitt{G) already belonging to the class X. The subsequent study is divided into two cases: the monolithic case and the non-monolithic case. In the monolithic case, as a rule, it is possible to prove, that Fitt(G) coincides with the monolith of a group; in the non-monolithic case Fitt{G) stands by a just infinite module. Therefore a tool in this study is the developing of properties of simple and just infinite modules, which were investigated in the previous chapters.
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Chapter 10 The Fitting Subgroup of Some Just non- X -Groups
In this chapter, we collect some of the most relevant properties of just non-X -groups we need in the sequel, especially those which concern the Fitting subgroup. Let G be a group. We recall that G is said to be monolithic if the intersection M of all its non-identity normal subgroups is non-identity: in this event M is called the monolith of G and will be denoted n(G). Evidently p.(G) is the unique minimal normal subgroup of G. Otherwise, if that intersection is identity, G is said to be non-monolithic. Clearly we have 10.1. Lemma Let X be a class of groups which is closed under taking subgroups and Cartesian products, that is, X is S and C-closed, (in particular, if Xis a variety ofgroups [RD 9]). Then a just non-X -group G is monolithic. Proof If S is the set of all non-identity normal subgroups of G and M = C\S, by Remak's theorem we can embed GIM'm the Cartesian product ri// e ,s GIH. Since G is a just non-A1-group and X is S and C-closed, it follows that GIM e X. But G . 10.2. Corollary (i) A just non-abelian group is monolithic. (ii) IfNc denotes the class of all nilpotent groups ofnilpotency class < c, then a just non-Nc-group is monolithic. 10.3. Lemma
Let X be a formation of groups and let G be a just non-X-group.
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110
Then G does not include two non-identity normal subgroups U and V such that
un v = < 1 >. Proof
Proceed as in the proof of Lemma 10.1.
10.4. Lemma [KO 2] Let G be a just non-FC-hypercentral nilpotent normal subgroup ofG, then L is abelian.
group. If L is a
Proof Suppose that L is non-abelian and choose in L a maximal G-invariant abelian subgroup A; thus A * L. Put ZIA = t^(LIA). Then ZIA is a non-identity G-invariant subgroup of LI A and ZIA f)FC(G/A) * < 1 > by Lemma 3.3. Let A * x0A e ZIA n FC(G/A) and define XIA = (< x0 >a )AIA. Then XIA is finitely generated abelian and the index \GIA : CGIA{XIA)\ is finite. Put YIA = CGIA(XIA). By the choice of A, X is non-abelian. Let x e X and consider the mapping *.
Actually p* is a ZK-endomorphism of A. Put Zi = f(L), so that Zi * < 1 >, and hence Zi < ^ . Obviously, J7Zi is FC-hypercentral and, since Z\ < CA(X), then YICA(X) is also FC-hypercentral. Suppose AICA(X) * < 1 >. Besides, Corollary 3.4 implies that A/CA(X) (\FC(YICA(X) *< 1 >. Let CA (A) * a d (A) e ^ / Q ( A ) 0 FC(YICA(X) and choose /7C^(i0 e Cy / C / 4 W (aQ(A)). Now aA = aft, where ft e C,i(X), and {a.Then F is abelian and either F is torsion-free or there exists a prime p such that F is p-elementary abelian. Furthermore, CQ (F) = F. Proof Let x,y e F, then there are the nilpotent normal subgroups Lx, Ly such that x e Lx, y e Ly. By the Fitting's theorem (see, for example, [RD 9, Theorem 2.18]), LxLy is nilpotent and so abelian by Lemma 10.4. Thus F is abelian. Let Tbe the periodic part of F. If T * < 1 >, then by Lemma 10.2 there exists a prime p such that T is a p-group. Suppose that 1? * < 1 >. Then T* T\ =Qi(T) = {x e T \ xP = 1>. Since GIT\ is FC-hypercentral, then by Corollary 3.4 (T/T{) f] FC(G/Ti) * < 1 >. In particular, TIT\ includes a finite non-identity G-invariant subgroup PIT\. Put HIT\ = CciT\(PIT\). Then H is a normal subgroup of finite index. If h e H and c € P\T\, then Ti = [cTuhTi] = [c,/j]ri and so [c,h] eTi.lt follows that [C^/J] = [c,K\p = 1Since c £ T\,cp * 1, and so £(//) * < 1 >. As other times, this yields that FC(G) * < 1 >, because \G : H\ is finite. This contradiction shows that T^ = < 1 >, that is, T = T\ is elementary abelian. Suppose now that F ± T. Then we may decompose F = A x r for some subgroup ^ (see, for example, [FL 1, Theorem 27.5]). It follows that Fp < A and, in particular, Fp fl T = < 1 >. Since F ± T,FP * < 1 >. This contradicts Lemma 10.2. Thus F = Tas claimed. For the last assertion, suppose that C = CG{F) * F. Since GIF is FC-hypercentral, by Corollary 3.4 we have that (OF) n FC{GIF) * 1. Let F*xFe (OF) n FC(GIF) and define XIF =< xF >G/F . Then XIF is central-by-finite. If RIF = C,(XIF), then 7? is a nilpotent normal subgroup of G, so i? < F. It follows that XIF is finite. Since F < ((X), by Schur's theorem (see, for example, [RD 9, Theorem 4.12]) the derived subgroup [X,X\ is finite. Therefore [X,X\ = < 1 >, andXbecomes abelian. Hence X < F, which is a contradiction. We immediately obtain the following consequences. 10.6.Corollary [KO 2] Let G be a just non-CC-group. IfFC(G) = < 1 > and Fitt(G) * < 1 >, then either Fitt(G) is a torsion-free abelian group or there exists a prime p such that Fitt(G) is an elementary abelian p - subgroup. Furthermore, CG(Fitt(G)) = Fitt(G). 10.7. Corollary [FdeGK 3] Let G be a just non-FC-group. If FC(G) = < 1 > and Fitt(G) * < 1 >, then either Fitt(G) is a torsion-free abelian group or there exists a prime p such that Fitt(G) is an elementary abelian p - subgroup.
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Moreover, CG{Fitt{G))
= Fitt{G).
In considering of just non-hypercentral groups, we obtain similar results. For the proof we shall need other auxiliary statements. 10.8. Lemma [KSU 3] Let G be an infinite just non-hypercentral-group. includes a finite normal subgroup, then G is finite.
If G
Proof Suppose G is infinite. Let A be a finite minimal normal subgroup of G. If A is not abelian, then A % CG(4) and so A n CG(A) = < 1 >. But, since G is infinite and G/CG(A) is finite, we have that CG(A) * < 1 >, which contradicts Lemma 10.3. Therefore A is abelian. Indeed, by minimality, A is elementary abelian. Since GIA is hypercentral but G is not, it turns out that A is exactly the hypercentral residual of G. By Robinson's Theorem 4.5, G splits over A, that is G = A X H for some subgroup H. Obviously this H is an infinite hypercentral subgroup and \H : CH{A)\ is infinite; in particular CH(A) * < 1 >. Since CH(A) is normal in H, C = CH{A) (1 £(//) * < 1 >• But C < C(G), so C(G) * < 1 >, a contradiction. 10.9. Theorem [KSU 3] Let G be an infinite just non-hypercentral group. If Fitt{G) * < 1 >, then either Fitt(G) is a torsion-free abelian group or there exists a prime p such that Fitt(G) is an elementary abelian p - subgroup. Moreover CG{Fitt{G)) = Fitt(G). Proof Suppose that FC{G) * < 1 >, pick l * j e FC{G) an put X = < x >G . If \x\ is finite, then X is finite too. Thus Lemma 10.8 shows that |x| is infinite. It follows that X includes a non-identity G-invariant torsion-free abelian subgroup A. Let r = ro(A). Given a prime p, we have that Ap * < 1 > and GIAP is hypercentral. Since \GIAP\ = pr, GIAP < (,r(GIAp). In other words [Gr,A] = [G,...,G,A] <AP. Since this holds for every prime p, r
\Gr,A\ < pi e¥Ap = < 1 >, and so A < £V(G) and gives G hypercentral. This contradiction shows that FC(G) =< 1 >, and then it suffices to apply Theorem 10.5. 10.10. Theorem [RW] Let G be a just non-(polycyclic-by-finite)-group. IfFitt G * < 1 >, then either Fitt(G) is a torsion-free abelian group or there exists a prime p such that Fitt(G) is an elementary abelian p - subgroup. Moreover CG(Fitt(G)) = Fitt(G). Proof Since G satisfies the maximal condition on normal subgroups, A = Fitt{G) is nilpotent. Suppose that B = [A,A] * < 1 >. Then GIB is polycyclic-by-finite and so AIB is finitely generated. Moreover, by a result due to
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Baer, the group A itself is finitely generated (see [RD 9, Corollary of Theorem 2.26]). Therefore G is polycyclic-by-finite, a contradiction which shows that A has to be abelian. Let Tbe the periodic part of A and suppose that T * < 1 >. By Lemma 10.3, there exists a prime p such that T is a p-group. Put T\ = Qi(T) so that T\ is an (infinite) elementary abelian p-group and GIT\ is polycyclic-by-finite. It follows that TIT\ is finite and in particular T is bounded. A celebrated result due to Priifer allows us to express T as a direct product of cyclic groups T =XASA < t% > (see, for example, [FL 1, Theorem 17.2]). Thus, if T * Tu then 1? * < 1 > and TIV is finite. Since V = T/Qi(T) is finite, we can deduce that Tis finite. Since this is impossible, T = T\. Suppose that T 4= A. We may decompose A = Tx U, for some subgroup U (see, for example, [FL 1, Theorem 27.5]). It follows that Ap < U, and, in particular, Ap is a non-identity torsion-free normal subgroup. Therefore Ap C\ T = < 1 >, which contradicts Lemma 10.3. This shows thaty4 = T. Finally, the assertion CG(A) = A can be proved proceeding as in the proof of Theorem 10.5.
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Chapter 11 Just non-Abelian Groups
Chronologically the class of just non-abelian groups was the first class of just non-A'-groups. The structure of soluble such groups was determined by M. F. Newman in his papers [NM 1] and [NM 2]. Making use of the results previously showed in the precedent chapters of this book, we can carry out this study more concisely. Let G be a just non-abelian group; by Corollary 10.2 G is monolithic. Thus let M = n(G) be the monolith of G. As mentioned above, we suppose that Fitt(G) * < 1 >. If M is not central in G, then £(G) = < 1 >, but M is abelian. Let A be a maximal abelian normal subgroup of G. Considering A as a Z//-module, where H = GIA is an abelian group, we can think of Mas a simple Z//-submodule of A. Suppose that M * A. If AIM has elements of finite order, we may choose in AIM a cyclic subgroup CIM of prime order. Note that C is a Z//-module since GIM is abelian, and so CIM is a simple Z//-module as well. By Corollary 4.3, there exists a non-identity G - invariant subgroup E < C such that C = M x E. In particular, MC\E =< 1 >. This contradicts Lemma 10.3 and so AIM is torsion-free. Suppose that M is an elementary abelian p-group, for some prime p. Then there exists a subgroup U such that A = Mx U (see, for example, [FL 1, Theorem 27.5]). Therefore < 1 > * Ap < U and so Ap C\ M = < 1 >, which contradicts Lemma 10.3. However, if M is torsion-free, then Corollary 4.4 and again Lemma 10.3 lead to another contradiction. Consequently A = M and so the monolith of G is a maximal abelian normal subgroup of G. Let L be a nilpotent normal subgroup of G; then f (I) = M. Suppose that L * M and choose an element x e DM, we can form the abelian group K = < x, M >. This is a normal subgroup of G since GIM is abelian. However M * K, which is impossible. Thus M = L and hence M = Fitt(G). We may apply the Robinson's result (Theorem 4.5) to obtain that G conjugately splits over M. On the other hand, if S is a complement to M in G, S = GIM is abelian and we may think of M as a simple ZS-module and apply
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Corollary 2.4 and Theorem 2.6. All this gives the case in which the monolith is not central, which is our next result. 11.1. Theorem [NM 1] Let G be a just non-abelian group with Fitt(G) * < 1 > and ((G) = < 1 >. Let M be the monolith ofG. Then: (1) M - Fitt(G) = CG(M) is the unique maximal abelian normal subgroup of G. (2)M=[G,G\. (3) G = M X S, S is abelian and any complement to M in G is conjugate to S. (4) The periodic part T = t(S) ofS is a locally cyclic group. (5) IfM is an elementary abelian p - subgroup for some prime p, then T is a pi - subgroup. (6) Ifro(G/M) is finite, then G is periodic. Obviously, these conditions above are also sufficient. Suppose now that the monolith M = ji(G) is central in G. In particular, ((G) * < 1 > and then G is nilpotent. Obviously M has to be cyclic of order p, for some prime p, and besides M = [G,G]. For each g e G, the mapping w e m a y transform A = EIC into a non-degenerate symplectic space over ¥p of countable dimension. By Proposition 10.3, A
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decomposes into an orthogonal direct sum of hyperbolic planes and the pre-image of each one of these hyperbolic planes corresponds to non-abelian subgroups of £ of order p3, which cuts each to other in C. As E is generated by all these pre-images, then the result follows. It is worth mentioning that, for uncountable groups, we have no analogy of Theorem 10.4 as the following example shows. 11.5. Example [HP 3] Given a prime p, let < a >,< b„ >,< c„ >,n e N, copies of the cyclic group of order p. Form B =X„ and C = n „ e N < c» >- Then C acts on the direct product < a > xB by the rules given by ac» = a,bc„" = b„a,bcn" = b „ , n,k e N , n =f= k.
Let E be the corresponding semidirect product. By construction £(£) = [£,£] = < a >. Since E/£(E) is an elementary abelian p-group, then E becomes an extraspecial p-group. Suppose we decompose £ in a direct products of groups Hi, A e A, of groups of order/?3 with < a > amalgamated. Since E is uncountable, so is A. However B is countable so that there exists a countable subset T c A such that B < Er =< H\ \ A e T >. This gives that < Hx \ A e AT > < CE(B) and, in particular, EICE{B) is countable. But CE(B) = < a > x B and £/(< a > xB) = C is uncountable. This contradiction shows that our assumption is not possible. The structure of uncountable extraspecial p-groups is very complicated and remains still almost unknown. A survey of the known results is reported in [TM 2, Section 3].
Chapter 12 Just non-Hypercentral Groups and Just non-Hypercentral Modules
The class of all nilpotent groups and the class of all hypercentral groups are very natural extensions of the class of all abelian groups. Thus in our study of just non-A'-groups, the consideration of just non-hypercentral groups and just non-nilpotent groups should be the next step. As in the previous chapters we are studying the just non-hypercentral groups G with Fitt(G) * < 1 > . Some information about this subgroup was obtained in Chapter 10. In particular, A = Fitt(G) is abelian, and, as usually, we can consider A as a ZH - module, where H = GIA is a hypercentral group. If £ is a non-identity G - invariant subgroup( or, using the module language, B is a non-zero ZH - submodule) of A, then GIB is hypercentral, in particular, a ZH -factor-module AIB is ZH hypercentral. Let R be a ring, G a group. An RG - module A is said to be a just non-hypercentral (more precisely, just non-RG-hypercentral) if every proper factor-module of A is RG - hypercentral, but A is not RG - hypercentral. Similarly, A is said to be just non-nilpotent, if A is not RG- nilpotent, but every proper factor-module ofA is RG - nilpotent. Thus, the study of just non-hypercentral groups naturally requires the necessity of study of just non-hypercentral modules. Note that the study of these modules leads us to some other results 12.1. Lemma Let R be a ring, G a group, A a just non-RG-hypercentral module, (p an non-zero RG - endomorphism ofA. Then
are the R submodules of A. If we suppose that Ker
, then AIKer
= 0. This means that Ker
. 12.2. Corollary Let R be a ring, G a group, A a just non-RG-hypercentral module. Then EndRGiA) has no zero-divisors. 12.3. Corollary Let D be a Dedekind domain, G a group, A a just non-DG-hypercentral module, I = AnnoG(A), CII the center of the factor-ring DGII. Then CII is an integral domain. Proof For each element x e C the mapping ix : a —• ax, a e A, is a DG endomorphism of A. Furthermore, the mapping <E>: x —• ix,x e C, obviously, it is a homomorphism of the ring C in the ring EndoaiA) and Kerd? = Annix}{A) = I. By Corollary 12.2 C//is an integral domain. 12.4. Corollary Let D be a Dedekind domain, G a group and A a just non-hypercentral DG - module. Then either A is D - torsion-free or AnnoiA) = P e Spec(D). Clearly, D < C, therefore DI{Dr\Annoa{A)) is an integral domain. It follows that either (D f] AnnDG{A)) = < 0 > or (D fl AnnoG(.A)) is a maximal ideal of D. 12.5. Corollary Let R be a ring, G a group, l * z e C(G), A a just non-hypercentral RG - module, CG(A) = < 1 >. Then CA(Z) = < 0 >. Proof
Since z e £(G), the mapping (p : a —• a(z- 1), a e A,
is an RG - endomorphism of A. By Lemma 12.1 Kercp - Ann^{z - 1) = CA(Z) = < 0 >. 12.6. Lemma Let R be a ring, G a group, A an RG- module, U an upper RG hypercenter ofA. IfB is a non-zero submodule ofU, then B C\ £RG(A) * < 0 >. This statement could be proved precisely in the same way as its group-theoretic analogy. Denote by IIRG(A) the RG - monolith of module A, that is the intersection of all non-zero submodules of A. An RG - module A is said to be an RG - monolithic, if JJ.RG{A) * < 0 >, and
Just non-Hypercentral Groups and Just non-Hypercentral Modules
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123
otherwise.
12.7. Lemma Let Rbe a ring, G a hypercentral group, A an RG - module. If A is just non-hypercentral andRG - monolithic, then A is a simple RG - module. Proof Let M = HRG(A); then M =£< 0 > and M is a simple RG - submodule. Furthermore, £RG(A) does not include M. Suppose that A *• M. A factor-module AIM is RG - hypercentral, so that £RQ{AIM) = CIM * < 0 >. We can assume that CQ(M) = < 1 >. Let 1 * z e f(G), then the mapping q> : c —* c(z - l ) , c e C, is an RG- endomorphism of C, so that Imcp = C(z - 1) and Ker
, that is C = Imcp. Since C(z - 1) < M, Im
. By Theorem 10.9 A is abelian. Lemma 12.7 yields that A is a minimal normal abelian subgroup, that is A = M. The same Theorem 10.9 proves that CG(.M) = M. Since Fitt(G) includes every normal abelian subgroup, then M i s also a maximal normal abelian subgroup of G. 12.9. Lemma Let F be afield, G a hypercentral group, x an element of infinite order of the center C,(G), A an FG - module such that CG(A) = < 1 >. If A is just non-FG-hypercentral and non-monolithic, then A is F < x > - torsion-free. Proof Put J = F < x >, then J is a principal ideal domain. We will consider A as JG - module. Suppose that A is not J - torsion-free. By Corollary 12.4 Annj(A) = P e Spec(f). There exists an irreducible polynomial J[x) e J such that P = Jf{x). Since C,JG{A) =< 0 >, f(x)±{x-\), in particular, J{x - 1) + Jf(x) = J. Since A is non-monolithic, it includes a proper non-zero FGsubmodule B. Since AIB is FG -hypercentral, CIB = C,FG{AIB) *< 0 >, that is C * B. If c e OB, then c(x - 1) e B. On the other hand, cj%x) = 0. The equation J(x - 1) + Jf{x) = J implies that c + B = B, a contradiction. This contradiction shows XhaXA is J - torsion-free. 12.10. Lemma Let F be afield, G a hypercentral group, A an FG - module such that CG{A) = < 1 >. If A is just non-FG-hypercentral and non-monolithic, then G is torsion-free. Proof Let T be the periodic part of G and suppose that T * < 1 >. Assume firstly that charF = p > 0. Let Tp be a Sylow p- subgroup of G. Suppose that
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Tp *< 1 >; then Tp n C(G) * < 1 >. Choose an element 1 * x e 7> n £(G). Since char F = p, the additive group ^ is an elementary abelian p - subgroup. If follows that the natural semidirect product A X< x > is nilpotent ( see, for example, [RD 10, Lemma 6.34]), in particular, CA(X) * < 0 >. This contradicts Corollary 12.5. Therefore T is a pi- subgroup. Choose a non-identity element y e C, (G) fl T. Since A is non-monolithic, it includes a proper non-zero FG submodule B. Then AIB is FG - hypercentral, so £FG(A/B) = OB * < 0 >. By Maschke's theorem (see, for example, [CUR 1, Theorem 10.8]) there is an F < y > - submodule E such that C = E © B, in particular, E(y - 1) < £. On the other hand, C(y - 1) < B, so that £(y - 1) < 5. It follows that E(y - 1) = < 0 >, that is E < CAM- However, this contradicts Corollary 12.5. If char F = 0, then we repeat the arguments of the previous paragraph, and obtain again a contradiction, which shows that T = < 0 >. 12.11. Corollary Let G be a just non-hypercentral group, Fitt(G) * < 1 >. Suppose that Fitt{G) is an elementary abelian p - subgroup for some prime p. Then GIFitt{G) is torsion-free. 12.12. Theorem Let F be afield, G a hypercentral group of finite 0 - rank, A a non-monolithic FG - module. If every proper factor-module of A is FG hypercentral, then A is itselfFG -hypercentral. Proof Suppose the contrary. Then A is just non-FG-hypercentral. We can assume that Co(A) = < 1 >. By Lemma 12.10 G is torsion-free. Choose a non-identity element x e £(G). Put J = F < x >, then J is a principal ideal domain with the infinite set Spec(J). By Lemma 12.9 A is J - torsion-free. Let 0 * b € A, B = bFG, n= {P \ P e Spec(J) and BP * B}. By Theorem 1.15 the set n is infinite. In particular, we can find a maximal ideal Pen such that P * J(x - 1). Since J is a principal ideal domain, there is an element y e J such that P = Jy. There are the elements u,v e J such that 1 = yu + (x - l)v. Since A is J - torsion-free, BP * < 0 >, thus AIBP is a FG - hypercentral module. By Lemma 12.6 CIBP = BIBP f) C, FG(A/BP) * < 0 >. For every element c e C\BP we have c(x-l)eBP. On the other hand, cy e BP, hence c + BP = cl + BP = cyu + c(x - l)v + BP = BP. This contradiction proves that A is an FG - hypercentral module. 12.13. Lemma Let R be a ring, G a group, A a just non-RG-hypercentral RG module. IfB, C are two non-zero RG - submodule , then BDC *< 0 >. Indeed, if we suppose that B f] C =< 0 >, then using Remak's theorem, we obtain the embedding, A < AIB © AIC, which proves that .4 is RG - hypercentral. Let D be a Dedekind domain, G a group, A a DG- module. Suppose that A is D
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- torsion-free. Put V = {B | B is a non-zero pure DG - submodule of A}, PDG(A) =
f)V
12.14. Lemma Let D be a Dedekind domain, G a hypercentral group, A a DG module which is D - torsion - free. Suppose that A is a just non-hypercentral module andPDG{A) * < 0 >. Then PDG(A) = A. Proof Suppose the contrary, let R = PDG{A) * A. Then AIR is D - torsion free. Furthermore, clearly R is a DG - submodule, so that AIR is .DG-hypercentral. Hence CIR = C,DG{AIR) * < 0 >. Let B be a non-zero DG - submodule of R. It follows from the choice of R that RIB is D - periodic. In other words, R is D irreducible. Choose an element c e C\R, then (c + R)DG = cD + R. Since A is just non-hypercentral, G * CG(R)- Corollary 4.4 shows that C includes a non-zero DG - submodule E such that E n R = < 0 >. But this contradicts Lemma 12.13. This contradiction proves the equality A = PDG(A). 12.15. Lemma Let D be a Dedekind domain, G a hypercentral group of finite 0 - rank, A a non-monolithic DG - module which is D - torsion -free. Suppose that A is just non-hypercentral module and the set Spec(D) is infinite. Then PDG(4) = . Proof Suppose that PDG(4) * < 0 >. Lemma 12.14 yields that ,4 = PDG(A). In other words, for every non-zero DG - submodule B of A the factor-module AIB is D - periodic. Let 1 * u e A, U = uDG,n = {P \ P e Spec(D) and UP * U). Theorem 1.15 shows the infiniteness of the set n. Let Pen:. Since UIUP is finitely generated DG - module then it includes a proper maximal DG submodule MplUP. Since AIMp is DG - hypercentral, UIMp C\ CDG(AIMP) * < 0 > by Lemma 12.6. It follows that Uco(DG) < Mp for each x e G, because UIMp is a simple DG - module. Since it is valid for every Pen, Um(DG) < f\Ps„Mp. If we suppose that {\P&CMP = < 0 >, then Ua>(DG) = < 0 >, that is U < £DG(A), that is impossible. Hence V = f]Pex MP *< 0 >. Then UIV = (uD + V)IV. Since AlV'xs, D - periodic, AnnD(u + F ) = / * < 0 > . We have / = Pi*1... P,kl (see, for example, [NW, Chapter 1, Theorem 1.4]). This means, that Y\D{UIV) = {Pi,... ,P,}. On the other hand, by the election of V we obtain that TVD(U/V) = n is infinite. This contradiction proves the equality PDG(A) = . 12.16. Lemma Let R be an integral domain, G a group, A an RG-hypercentral module Suppose B is a non-zero RG-submodule of A which is R-torsion-free.
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Then B/(B n £RG(A)) Proof
is R-torsion-free.
Put C = CRG{A). By Lemma 12.6, B f~l C * < 0 >. Let T/(B D C) be the «
- periodic part of B/(B DC). If 6 e T, then tyefiflC for some 0 * >> e /?. Clearly r is an 7?(7-submodule of A. Given g e G,weputZ>i = Z>(g- l).Then b\y = &fe- l)y = by{g- 1) = 6y, so (b\ - b)y = 0. Since B is .R-torsion-free, b\ = b. Hence b e Bf)C, and so
r=finc. 12.17. Corollary
Le/ < 0 > = Co < Ci < ...C„ < C a+ i < ...
Cr=A
be the upper RG-central series of A. If A is R-torsion-free, then Ca is an R-pure submodule of A for every a < y. 12.18. Theorem Let D be a Dedekind domain with the infinite set Spec(D), G a hypercentral group of finite 0 - rank, A a non-monolithic DG - module which is D - torsion-free. If every proper factor-module of A is DG - hypercentral, then A is DG - hypercentral. Proof Suppose the contrary. Thenv! is a just non-hypercentral module. Lemma 12.15 yields that PDG(A) =< 0 >. Let F be the field of fractions for the ring D, E = A ®D F. We can consider E as an FG- module. Let U be a non-zero FG submodule of E, B=Uf]A. Then A/B = A/(A nU) = (A + U)IU. Hence A/B is D - torsion-free. Furthermore, B *< 0 > because E is an essential extension of A. It follows that A/B is DG - hypercentral. Let B = Bo , E is a non-monolithic FG - module. So we can use Theorem 12.12. Corollary 12.4 , Lemma 12.7, Theorems 12.12 and 12.18 directly imply
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12.19. Theorem Let D be a Dedekind domain with the infinite set Spec{D), G a hypercentral group of finite 0 - rank, A a non-simple DG - module. If every proper factor-module ofA is DG - hypercentral, then A is DG - hypercentral too. The above statements are just simple translations into the module language of the main statements of [KSU 3] 12.20. Corollary Let D be a Dedekind domain with the infinite set Spec(D), G a hypercentral group of finite 0 - rank, A a noetherian DG - module. If A is not DG - hypercentral, then A includes a maximal DG - submodule M such that the (simple) factor-module AIM is not DG - central. Proof Let M = {B \ B is a DG-submodule of A such that AIB is not Z)G-hypercentral}. Since < 0 > e M, M * 0. Since A is a noetherian DG module, the set M contains a maximal element M. If we assume that M is not a maximal DG - submodule, then AIM is not a simple DG- module. But in this case Theorem 12.19 yields that AIM is DG - hypercentral. This contradicts the choice of M and shows that M is a maximal DG - submodule of A. 12.21. Corollary Let D be a Dedekind domain with the infinite set Spec(D), G a hypercentral group of finite 0 - rank, A a noetherian DG- module. If every simple factor-module of A is DG-central, then A is a DG - nilpotent module. Now we can prove some results about just non-hypercentral groups. 12.22. Theorem [KSU 3] Let G be a non-monolithic group in which every proper-factor is a hypercentral group of finite 0-rank. IfFitt(G) * < 1 >, then G is hypercentral. Proof Put A = Fitt(G), and suppose that G is not hypercentral, i.e. G is just non-hypercentral. By Theorem 10.9 either .4 is an elementary abelian p-group, for a certain prime p, or A is a torsion-free abelian group. Furthermore, CQ(A) = A. The factor-group H = GIA is hypercentral. If A is an elementary abelian psubgroup, then A is an ¥PH - hypercentral module by Theorem 12.12. If A is an abelian torsion-free subgroup, then we can apply Theorem 12.18. 12.23. Corollary [KSU 3] Let G be a non-monolithic group with Fitt(G) * < 1 >. If every proper factor-group of G is a periodic hypercentral, then G is hypercentral. 12.24. Corollary [FdeG 2] Let G be a non-monolithic periodic soluble group. If every proper-factor is nilpotent, then G is nilpotent.
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In connection with Theorem 12.22 there appears a question about the existence of the non-monolithic soluble just non-hypercentral (respectively just non-nilpotent) groups. The following simple example provides the positive answer. 12.25. Example Let A = Q2, = {-f \ m,n e Z, w * 0 and n is odd } be the additive group of 2' - adic fractions, Po = {pk | k e N} the set of all odd primes. For every pk e Po the mapping k = p^a, a e A, is an automorphism of A. Let G be the natural semidirect product of A on X ^ N < • It is not hard to prove that if B is a non-zero G - invariant subgroup of A, then the index \A : B\ is finite, moreover \A : B\ = 2' for some ( £ N . Let U and V be non-zero G - invariant subgroups of ,4 such that V < [/and \UIV\ = 2. If u e U\V, then 2w e V. We have now u(. It follows that every proper factor-group of G is nilpotent. Finally, obviously, f]mN(2')A = < 0 >, therefore the group G is non-monolithic. The results above are concerned with the non-monolithic case. Now we are dealing with the monolithic one. 12.26. Theorem [KSU 3] Let G be a monolithic group with Fitt(G) * < 1 >. Then G is just non-hypercentral if and only if the following holds: (i) G = M X H where M = Fitt{G) is the monolith of G {in particular M is abelian), H is a hypercentral subgroup. (ii) M = CG{M) andH = NG(H). (Hi) All complements to Mare conjugate in G. (iv) The periodic part Tof£,(K) is locally cyclic. (v) If for a prime p M is an elementary abelian p-group, then T is a plsubgroup. Proof Let M be the monolith of G; then M < Fitt(G) and so M is abelian. By Corollary 12.8 Fitt(G) = M is a maximal normal abelian subgroup of G, so that M = CG(M). Since G is not hypercentral, then f (G) = < 1 >, and, in particular, £(G) fl M = < 1 >. By theorem 4.5, there exists a subgroup H of M such that G = M\ H, and, moreover, any complement to M in G is conjugate with H. Since Mis the unique minimal normal subgroup of G, H = NQ(H). Put C = £(//) and let T be the periodic part of C. Then assertions (iv) and (v) follow at once from Theorems 3.1 and 3.2. The converse is immediate. In a similar way, but making use of Corollary 1.16 and Theorem 3.1 in the
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appropriate place, we may obtain the following 12.27. Corollary [KSU 3] Let G be a monolithic group with Fitt(G) * < 1 >. If G is non-hypercentral and every proper factor-group of G is a hypercentral group of finite 0 - rank, then (i) G = M \ H, where M = Fitt(G) is the monolith of G (in particular M is abeliari), H is a hypercentral subgroup of finite 0 - rank; (ii) M = CG(M) andH = NG(H); (Hi) all complements to Mare conjugate in G; (iv) M is an elementary abelian p - subgroup for some prime p; (v) C,(H) is a locally cyclic pi - subgroup. To complete these results it is worth mentioning that simple Z//-modules over hypercentral groups has been already considered in Theorems 3.1 and 3.2. 12.28. Corollary [FdeG 2] Let G be a periodic soluble group. If G is a monolithic just non-nilpotent group, then (0 G = M\H; (ii) Mis a minimal normal subgroup ofG, M = CG(M); (Hi) M is an elementary abelian p - subgroup for some prime p; (iv) H = NG(H) is a nilpotent pi - subgroup. 12.29. Corollary [FdeG 2] Let G be a soluble non-nilpotent group. If G is a just non-Nc-group, then (i) G = M\H. (ii) M is a minimal normal subgroup ofG, M = CG{M)', (Hi) H = NQ(H) is a nilpotent subgroup of class < c; (iv) all complements to Mare conjugate in G. Indeed, by Corollary 10.2 the group G is monolithic, and we can use Theorem 12.26. And, finally, the nilpotent just non-Nc-groups are described by the following result. 12.30. Theorem [FdeG 2] A nilpotent group G is a just non-Nc -group if and only if there exists a prime p such that £(G) is a locally cyclic p-group and \yc+l(G)\=p. Proof Let G be a nilpotent just non-Nc - group and put C = £(G). By Lemma 10.3, n(G) = {p} for some prime p. We claim that C is periodic. Otherwise, let z e C be an element of infinite order. For each n e N z" * 1, and so the factor-group CI < z" > is an Nc - group. Since p| n € N < z" > = < 1 > by Remak's theorem, we have G
, and Wilson's results require rather special techniques, we simply quote the most fundamental results of the mentioned paper omitting their proofs. 14.5. Theorem [WJ 2] Let G be a just infinite group with HP(G) = < 1 >. IfH is a subnormal subgroup of G, then there exists a subnormal subgroup C such that < H,C > = Hx C and the index \G : HC\ is finite. 14.6. Corollary [WJ 2] Let G be a just infinite group with HP{G) =< 1 >. If H and K are two subnormal subgroups ofG, then < H,K > is subnormal too. In particular, the set Ssn(G) of all subnormal subgroups ofG is a sublattice of the lattice of all subgroups ofG. IfH,K
e SS„(G), then we write H~ K if\H : Hf\K\\K
: Hf)K\ is finite.
Obviously ~ is an equivalence relation. Moreover, ~ is a congruence on Ss„(G).
The factor - lattice C(G) ofSs„(G) lattice ofG.
by the congruence ~ is called the structure
14.7. Theorem [WJ 2] Let G be a just infinite group with HP{G) = < 1 >. If the lattice C(G) is infinite, then it is isomorphic with the lattice of the closed-and-open subset of the Cantor's ternary set. If C(G) is finite, then G satisfies the maximal condition for subnormal subgroups. The most near to finite groups are polycyclic-by-finite and Chernikov groups. Just non- (polycyclic-by-finite) groups will be considered in the following chapter. Just non-Chemikov groups we shall consider now. Moreover, some general situation will be considered here. A group G is called hyperfinite, if G possesses an ascending series of normal subgroups with finite factors. Clearly, every Chernikov group is hyperfinite, every periodic soluble-by-finite group of finite abelian section rank (in particular, of finite special rank) is hyperfinite. Note also that hyperfinite groups are exactly the periodic FC hypercentral groups. Also it is obvious, that the class of all hyperfinite groups is a
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149
formation. Now we will consider just non-hyperfinite groups. The study of such groups needs to be split on the following two cases: the FC center is non-identity and the FC - center is identity. 14.8. Theorem [KSU 4] Let G be a group with FC(G) *< 1 >. Then G is a just non-hyperfinite group if and only if G includes a normal abelian subgroup A satisfying the following conditions: (0 A is a torsion-free subgroup of finite 0 - rank; (ii)A = CG(A); (iii) A isX - irreducible in G; (iv) GIA is finite. Proof Let 1 * b e FC(G),B = < b >G ; then \G : CG(B)\ is finite and B is central-by-finite. By Schur's theorem (see, for example, [RD 9, Theorem 4.12]) [B,B] is finite. Since G does not include finite normal subgroups, it follows that [B,B] = < 1 >, i.e. B is abelian. In this case the set T of all elements of B, having finite orders, is a subgroup of B. Since B is finitely generated, the subgroup T is finite. Clearly T is normal in G, and again T = < 1 >. In other words, B is a free abelian subgroup of finite rank. Put C = CG(B). Since GIB is hyperfinite, [C,C] is periodic (see, for example, [RD 9, Corollary to Theorem 4,12]). However, in this case [C,C] V\B = < 1 >, which contradicts Lemma 10.3. This means that [C,C] = < 1 >, i.e. C is abelian. By the same reason, C is torsion-free. Let A be a maximal normal abelian subgroup of G including C. Using the previous arguments, we can prove that CG(A) is abelian and torsion-free. It follows that CQ(A) = A. Since CG{B) = C . Let 1 * e e A n FC(G),E = < e >G . Since GIE is hyperfinite, AIE is periodic. This means that A has finite 0 - rank and A is Z- irreducible in G. Conversely, assume that G includes a normal abelian subgroup A satisfying the conditions (i) - (iv), H is a normal non-identity subgroup of G. Since A = CQ(A), A fl H * < 1 >. By (iii) AI(A n H) is periodic, so GI(A fl FT) is periodic and abelian-by-finite. Clearly, such groups are hyperfinite. 14.9. Corollary Let G be a group with C, (G) * < 1 >. Then G is a just non-hyperfinite group if and only ifG is a torsion-free locally cyclic group. 14.10. Corollary Let G be a group with FC(G) i= < 1 >. Then G is a just non-Chernikov group if and only if G includes a normal abelian subgroup A satisfying the following conditions: (i) A is torsion-free minimax subgroup; (ii)A = CG(A);
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{Hi) A is Z- irreducible in G; (iv) GIA is finite. These assertion are contained in Theorems B and C of paper [FdeG 1]. Now we will consider just non-hyperfinite groups with identity FC - center. 14.11. Proposition [KSU 4] Let G be a just-non-hyperfinitegroup 'with FC{G) = < 1 >. IfFitt(G) * < 1 >, then either Fitt(G) is torsion-free abelian or Fitt(G) is an elementary abelian p - subgroup for some prime p. Moreover, Ca(Fitt(G)) = Fitt{G). In fact, since FC{G) = < 1 >, G is a just non-FC-hypercentral group, and we may apply Theorem 10.5. 14.12. Lemma [KSU 4] Let G be a just non-hyperfinite group with FC{G) = < 1 >, A a non-identity normal abelian subgroup ofG,FICa{A) a finite normal subgroup of G/CG(A).
ThenCA(F)
=
.
Proof Put Q = GICG{A),HICG{A) = CQ{FICG{A)); then H is a normal subgroup of finite index. Suppose the contrary, let C = CA(F) * < 1 >. Let 1 * g e F\CG(A). Consider the mapping . It proves that CA(V) * < 1 >, which contradicts Lemma 14.12. This contradiction shows that Opi(GIA) is finite. But in this case GIA is finite by Corollary 14.14, and hence FC(G) i= < 1 >. This final contradiction proves that M(A) = 0. 14.16. Lemma [KSU 4] Let G be a just non-hyperfinite group, A = Fitt(G) * < 1 > and suppose that A is torsion-free. Then FC(G) * < 1 >.
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Proof Suppose the contrary, let FC(G) = < 1 >. Proposition 14.11 yields that A is abelian and CG(A) = A. If we assume that ro(A) is finite, then G/A is isomorphic to some periodic subgroup of GLr(Q), where r = ro(A). However periodic subgroups of GLr(Q) arefinite(see, for example, [WB, Theorem 9.33]), so G/A is finite and hence FC(G) * < 1 >. This contradiction shows that ro(A) is infinite. Let K* = KJA be a finite normal subgroup of G* = G/A. FutE = A ® z . We may consider £ as a QG* - module. Moreover, E is a simple QG* - module, because A is Z - irreducible in G. Let 1 * u s A and U a QK* - submodule, generated by u. Since K* is finite, dimq U is finite, so that U includes a simple QK* - submodule V. Lemma 3.5 assures the existence of a subset S such that (in additive notations) E = ®xeS Vx. Since E is a Z - essential extension of A, EIA is periodic. It follows that and V/(V(~)A) is periodic, in particular, Vf\A * < 1 >. Let 1 * w e Vf]A,W = <w>K; then W and Fitt(G) =A± < 1 >. Then G is a just non-Chernikov group if and only if G satisfies the following conditions: (i)A is an infinite elementary abelianp - subgroup for some prime p; (ii) A is the monolith ofG (i.e. A is the unique minimal normal subgroup ofG); (iii)A = CG(A); (iv) G = A \ H where H is a Chernikov subgroup; (v) the complements to A in G are conjugate with H; (vi) ifS = Socat(H), then S is a pi-subgroup, in particular, the divisible part of H is a pi- subgroup; (vii) S includes a subgroup Q such that SIQ is locally cyclic and Coreu(Q) = < 1 >. For the case of a soluble group G the statements (i) - (iv) and (vi) have been proved in [FdeG 1]. The following question arises in connection with the results of this Chapter and Chapter 12 . Question 6 Describe the structure of a group G with Fitt(G) * < 1 >, every proper factor-group of which is a FC - hypercentral group of finite 0 - rank.
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Chapter 15 Just non-(Polycyclic-by-Finite) Groups
The classes V of all polycyclic-by-finite groups and C of all Chernikov groups are the most near extensions of the class T of all finite groups and they have been studied intensively by several authors. Since these groups are finite extensions of soluble groups with the two main chain conditions Max and Min respectively (see [RD 9, 3.1]), the classes V and C are dual to each other in some sense. Also V fl C = T. In the previous chapter we have already considered just non-Chernikov groups. This chapter will be devoted to just non-(polycyclic-by-finite) groups. J. R. J. Groves [GJ 1] initiated the study of these groups. More specifically, he considered the metanilpotent case. The description of soluble just non-polycyclic groups has been obtained by D. J. S. Robinson and J. S. Wilson [RW]. We have already remarked, that this paper [RW], saturated with new ideas, constructions, and results, has played a great role in the theory of just non-A'-groups. In Chapter 8 we have already exposed some results of [RW]. The current chapter exposes other results of [RW] devoted to just non-polycyclic groups. We start with some elementary properties of just non-(polycyclic-by-finite) groups, which we shall freely use in what follows. 15.1. Lemma [RW] (i) Every finitely generated soluble non-polycyclic group has a just non-polycyclic factor-group. (ii) Every just non-{polycyclic-by-finite) group satisfies Max-n, the maximal condition on normal subgroups, and is finitely generated. (Hi) IfG is a just non-(polycyclic-by-finite) group and Fitt(G) * < 1 >, then G is abelian-by-polycyclic-by-finite. The last statement follows from Theorem 10.10.
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We need the following ring-theoretical lemma. 15.2. Lemma [RW] Let R be a commutative ring, A an R-module having a finite composition series, S = SOCR(A). If A/S and S have no isomorphic simple submodules, then A = S. Proof Suppose that A =t= S. Let BIS be a simple R - submodule of A/S. Suppose that S = Si © ... © S„ where St is a simple /^-submodule of A, 1 < /' < n, and put I = Annit(B/S), Ij = AnnR{Sj), 1 <j < n. By Chinese Remainder Theorem (see, for example [LS, Chapter 11, Section 2]) there is an element x e R such that x e I, x e I + Ij for each j , 1 <j. Suppose that Kerfx n S * < 0 >. Then S fl Kerfx includes a simple ^-submodule U. Since S = ©! e AnnR(U), we can write Ufx = Ux= U(l+y) = U+Uy= U. On the other hand, t/ < #er& so Ifc = < 0 >. This contradiction shows that Kerfx fl S = < 0 >. Since 5/5 is a simple /{-submodule, B = S(B Kerfx and £er/i is a simple i?-submodule. However in this case Kerfx < SOCR(A) = 5. This contradiction shows that .4 = S. 15.3. Proposition [RW] Let G be a just non-(polycyclic-by-finite) group, A = Fitt(G), H = GIA. Then A is a just infinite ZH - module. Conversely, let H be a polycyclic-by-finite group, A a just infinite ZH - module such that CH{A) — < 1 >. If A is not Z-finitely generated then every extension G of A by H is a just non-(polycyclic-by-fmite) group such that Fitt{G) = A. Proof Let G be a just non-(polycyclic-by-finite) group. By Theorem 10.10 either A is an elementary abelian p-subgroup for some prime p, or A is torsion-free abelian. Let us begin with the periodic case, so suppose that A is an elementary abelian p-group, p is prime. Given a non-identity G-invariant subgroup B of A, we have that AIB is finite since GIB is polycyclic-by-finite. Thus the condition (JI 1) is clearly satisfied and so it suffices to check the condition (JI 2). Let Tia(A) = {B I B is a non-identity G - invariant subgroup of A}, and put C = f]Ha(4). Suppose that C * < 1 >. By the election, C is a simple WpH - submodule of A. But these modules are finite [RJ 1]. This contradiction shows that p | W o ( ^ ) = < 1 >. Therefore we have just checked (JI 2) and hence A is even a just infinite Fp/Z-module in this case. Let now A is a torsion-free and abelian. We split the proof in two
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complementary cases assuming first that A is Z-irreducible. If B e HG(A), then GIB is polycyclic-by-finite. Since AIB is periodic, AIB is finite. Suppose again that C * < 1 >. Corollary 1.19 yields that C is an elementary abelian p-subgroup for some prime p, what is impossible because A is torsion-free. So P | 7 7 G 0 4 ) = < 1 > and A is a just infinite 7LH - module. Therefore we may assume that A is not Zirreducible. Then A includes a G - invariant subgroup B such that AIB is not periodic. Let E\IB be the periodic part of AIB, then E\IB is finite and AIE\ is a free abelian group of finite 0 - rank. Suppose that E\ is also not Z - irreducible. Then E\ includes a G - invariant subgroup Ei * £i such that £i/£2 is a free abelian group of finite 0 - rank. Otherwise, we may iterate the above construction and build up an infinite (proper) descending chain A > E\ > E2 > ... > E( > ... such that A/E, is a free abelian group of finite 0 - rank and ro(A/Ej) < ro(AIEi+i) for any i e N . Letp be a prime. Then \{AIEi)l{AIE,)p\ < \{AIEt+x)l{AIEi+\)p\ for every n e N. It follows that AIAP is infinite. Ap * < 1 > since A is torsion-free. Hence GIAP is polycyclic-by-finite, and therefore AIAP is finite. This contradiction shows our claim and then A includes a Z-irreducible Z//-submodule U, and All] is free abelian of finite rank. By the arguments above, U also becomes a just infinite Z//-module; in particular, U is a finitely generated Z//-module. By Corollary 1.8 there is a finite set n of primes such that U e A(Z,Tt). Suppose that q £ n. Lemma 1.6 implies the equality rz(U) = dim¥q{UIUq). Since Uq * < \ >, GIW is polycyclic-by-finite, in particular, UIUq is finite. It follows that r%(ll) = r0(U) is finite. Since All! is finitely generated, and ro(A) is finite too. In this case A has a finite series of 7LH - submodules U = Uo < Ui < ... < U„ = A such that Ui is a pure subgroup of A and a Z/f - module C/,7C/,--i is Z - irreducible for any ;', 1 < i < «. It follows that G/CG(U) and G/CG(t/,/f/,-i) are abelian-by-finite (see, for example [WB, Lemma 3.5]). Put
L = cG(£/) n cG(UiiUo) n... n c G (t/ n /f/„-i). Then G/i is abelian-by-finite by Remak's Theorem. Moreover, LIA is nilpotent (see, for example, [KW, Theorem l.C.l]). By construction L is nilpotent itself, so that L < Fitt{G) = A. In other words, GIA is abelian-by-finite. Let K be a normal subgroup of finite index such that H\ = KIA is abelian. We consider A as Z//j module. Put V = A z Q , Vo = t/o ®z Q. Since £/o is Z- irreducible, V0 is a simple QH - module. By Proposition 3.6 Vo includes a simple Q//i - submodule W such that VQ = Wx\ © ... © fFx, for some elements x\,...,xt e H. Let
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D = i/o fl W. Obviously D *< 1 >. Assume that D is finitely generated as subgroup. Choose the elements y\,...,yt such that x\ =y\A,... ,xt — ytA. Then Dy'... Dy' is a Z// - submodule which is a finitely generated as a subgroup. Since Uo is Z - irreducible, UolDy'...Dy' is finite. Hence f/o is finitely generated as a subgroup, which is impossible. This contradiction shows that the subgroup D is not finitely generated. Let S/V0 be a simple Q//i - submodule of V/V0, R = SC\A. Then R/U0 is a Z irreducible QHi - submodule of A/UQ. Since A/Uo is finitely generated as a subgroup, Z//i - modules DXi and 7?/t/o are not isomorphic. Therefore CHl (DXl) * CHl (R/Uo), \ and G be an extension of ^ by H. Let Xbe a non-identity normal subgroup of G. If we assume that Xf\A = < 1 >, then X = X4A4 is isomorphic with some subgroup of CH{A) = < 1 >. So XC\A * < 1 >. It follows that/l/(Xn^) is finite, therefore G/(XC\A) and also G/Xis polycyclic-by-finite. Finally, let A] = Fitt{G). Then A , H = GIA. If A is not periodic, then (i) A is a minimax abelian torsion-free subgroup; (ii) the ZH- module A is just infinite; (Hi) H is abelian-by-finite; (iv) there exists a free abelian subgroup X of finite 0 - rank such that XC\A = < 1 > and the index \G : XA\ is finite. Moreover, if G splits over A, the complements of A fall into finitely many conjugate classes. Proof By Theorem 10.10 A is a torsion-free abelian subgroup. By Proposition 15.3 A satisfies (ii), in particular, A is a finitely generated Z// - module. By Corollary 1.8 there exists a finite set of primes n such that .4 € A(Z,n). Choose a prime q £ n; then by Lemma 1.6 r%(A) = ro(A) = dimtq(AIAi). Since A is torsion-free thus Aq *< 1 >, so GIAq is polycyclic-by-finite. In particular, AIAq is finite, and therefore ro(A) is finite. Together with ,4 e A(l, / / = GIA. Suppose that A is periodic andFC(H) is infinite. Then (/) A is an elementary abelian p-subgroup for some prime p; (ii) H includes a normal torsion-free abelian finitely generated subgroup L of finite index; (Hi) there is an element 1 =£ x e L such that A is¥p < x > - torsion-free and ¥p < x >-minimax; (iv) G includes a free abelian subgroup X of finite 0 - rank such that X DA = < 1 > and the index \G : XA\ is finite. Moreover, if G splits over A, the complements ofA fall intofinitelymany conjugacy classes. Proof (i) follows from Theorem 10.10. Put F = FC(H). Then F is a finitely generated FC - group, so F is central-by-finite. Moreover, the index \H : Cn(F)\ is finite. If C = Q(F), then C is finitely generated abelian subgroup and there is a number t e N such that D = C is torsion-free. Since D < FC(H), the subgroup K = CH(D) has finite index. By Proposition 15.3 A is a just infinite"LH- module. We may apply Proposition 6.9 to deduce that A includes an ¥PK - submodule B such that AIB is a just infinite FPK - module. Moreover, if T is a transversal to K in H, then A/By is a just infinite ¥PK - module for every y £ T and A < ®y&TAIBy. Suppose that£>/C/)C4/B) is finite. Then by DICD(AIBy) = D/y-lCD(A/B)y = (y-lDy)/(y~iCD(A/B)y) s D/CD(A/B), DICo(AIBy) is finite too any y € T. Proposition KICK(A) < XyeTKJCK(A/By). It follows that
6.9
implies
that
D < Xy&T(DCK(A/By)/CK(A/By)) = XyeTDICD(AIBy). In particular, D is finite because CD(A) = < 1 > by Theorem 10.10. This contradiction shows that DICD(AIB) contains elements of infinite order. Since D < C(K), proposition 6.18 and Corollary 7.19 imply that K/CK(A/B) is abelian-by-finite. From the embedding K/CK(A) < Xyt=TK/y~lCK(A/B)y we can see that KJCK(A) itself is abelian-by-finite. By Theorem 10.10 CK(A) = < 1 >, therefore K and H are abelian-by-finite. Fix a normal torsion-free abelian subgroup L of//having in //finite index, so that we have just proved (ii). Proceeding as above, we may conclude that A includes an ¥PL - submodule U such that Al'U is a just infinite FPL - module. Moreover, if S is a transversal to L in H, then A/Uy is a just infinite ¥PL - module for each y e S and A < 0 SA/Uy. In the same way LICL(AIU) is not periodic. Hence L contains an element x\ such that < xi > C\CL(AIU) =< 1 >. By Corollary 7.20 AIU is ¥P < x\ > - torsion-free
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and ¥p <x\ > - minimax. Let S = {yi,... ,yi}. We proceed by induction on the number t of summands, starting with the case t = 1. Let X2 be an element of L such that <x2 > nCL(A/Uyi) = < 1 >. If < xi > f]CL(A/Uyi) = < 1 >, then AlUyi is ¥p < x\ > - torsion-free and ¥p < x\ > - minimax. In this case AIU ® AIUy\ become also ¥p < x\ > - torsion-free and F^ < x\ > - minimax. In this case we put x = xi. If < x\ > rvCi.(A/Uyi) * < 1 > but < X2 > p[Ci{AIU) = < 1 >, then put x = xi. Again AIU © A/Uyi is Fp < x > - torsion-free and Fp < x > minimax. Thus we have reduced the proof of (iii) to the case in which < x\ > f] CiiAIUyi) * < 1 > and < x2 > P[CL(AIU) * < 1 >. Then there are numbers ki,k2 such that x\' e CL(AIUyx), and x*2 e CL(AIU). Put x = x\'xk2\ Then x acts on AIU as x{' does, and on AIUy\ as x22 does. It follows that AIU © AlUy i is ¥p < x > - torsion-free and ¥p < x > - minimax. The general case follows inductively. As in Theorem 15.5, the assertion (iv) follows from [RD 13, Theorem 5]. To complete the periodic case, note that, since G is not polycyclic-by-finite, the complementary case to Theorem 15.5 is the case where FC(G) = < 1 >. To deal with this case, we shall need an auxiliary result which allows us to reduce our study to a subgroup of finite index. For our convenience, we shall say that a just non-polycyclic group G with Fitt{G) * < 1 > is called primitive if the factor-group GIFitt{G) is primitive in the sense given in Chapter 8. 15.6. Lemma [RW] Let G be a just non-(polycyclic-by-finite) group with < 1 > * Fitt(G) periodic. Then G includes a subgroup H of finite index such that H is normal in HG and H has a primitive just non-polycyclic factor-group HIK. Proof We choose a section H\IK\ of G satisfying the following three conditions: (a)|G : Hx\ is finite; (b) H\IK\ is a just non-polycyclic group; (c) 0-rank of the Fitting factor-group of H\IK\ (that is O-rank of the group {H\IK\)IFitt{H\IK\)) is minimal under conditions (a) and (b). Let H2 = CoreG{H\). Then H2I(H2P\K\) is a finitely generated soluble-by-finite group which is not polycyclic and so by Lemma 15.1 H2/(H2 so that Kf\D < CW{D) = < 1 >. This in turn implies that [K,D] = < 1 > and K < CW(D) = < 1 >. It follows that K n Hy * < 1 > for every y e Y, thus each Hyl(Hy fl K) is polycyclic-by-finite, and consequently WIK is polycyclic-by-finite too. 15.8. Proposition [RW] Let G be a group, H a subgroup of G, K a normal subgroup ofH. Write a for the canonical permutation representation ofG on the set of right cosets of H in G (so that Go = G/Coreo(H)). Let W be the unrestrictedpermutational wreath product: W = (HIK) WR (Go). Then there is a homomorphism y/ : G —* W satisfying Kery/ = Corea(K). Moreover, ifD is the base group of the wreath product then D(Gy) = W, Df] Gy/ = CoreG(Hyi), and the latter projects onto the canonical direct factors of the base group of (KCoreG(H)/K) WR Go. Proof Let Z be the set of right cosets of H in G and regard the base group of a wreath product X WR Go as the group of all mappings from S to X. We also write 8 for the "diagonal" mapping from .Yto this base group (so that if x e X, then x8 maps each element of £ to x); thus the subgroup < X5, Go > of X WR Go is the direct product of X8 and Go. A right transversal to H in G may be regarded as a function 1: 2 —• G mapping each coset to its representative, and so as an element of G WR Go. We form the mapping g - • i(g8)(go)rl from G to G WR Go; this
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is already a monomorphism and, as is observed in paper [CKK ], its image lies in H WR Ga. Thus we have an embedding q> : G —• H WR Ga. Define y/: G —• (H/K) WR Ga be the composite of cp and the canonical homomorphism £: H WR Ga — (H/K) WR Ga. Let g e G. Evidently gy/ lies in the base group D of (H/K) WR Ga if and only if g(p lies in the base group of H WR Ga, and from the definition of
. Theorem 8.5 shows that the statements (iii), (iv), (v) hold We finish with the following interesting remark. Let G be a group, H a normal subgroup of G. We say that H is nearly complemented in G or G is nearly splits over H if there is a subgroup X such that HP\X = < 1 > and the index \G : HX\ is finite. With this terminology, Theorem 15.4 asserts that if G is a just non-(polycyclic-by-finite) group with < 1 > * Fitt(G) torsion-free, then G nearly splits over Fitt(G). When A = Fitt(G) is periodic, we have only proved a partial version of this, namely G also nearly splits over Fitt(G) provided FC(GIA) is infinite. Actually, this is a fairly good result since the following result has been proved. 15.12. Theorem [RW] There exists a primitive just non-polycyclic group G which does not nearly split over its Fitting subgroup. The proof of the above result heavily depends on homological techniques exceeding the scope of this book, which is the reason for omitting it.
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Chapter 16 Just non-CC-Groups and Related Classes
Now we want to consider just non-Af-groups, where A" is a certain class of groups which includes both classes: the class T of all finite groups and the class A of all abelian groups. Specifically, we will consider the following candidates for X: central-by-finite groups, finite-by-abelian groups, FC-groups, and CC-groups. Just non-(central-by-finite) and just non-(finite-by-abelian) groups have been investigated in the remarkable paper of D. J. S. Robinson and Z. Zheng [RZ ]. While in [RW ] the key condition on the respectively group ring was to be noetherian, in the case of central-by-finite or finite-by-abelian group H the group ring Z// can not be noetherian. In this case the following fact is defining: the group H contains many almost central elements. Since central-by-finite and finite-by-abelian groups are contained in the wider class of FC - groups, the following natural step was the study of just non-FC-groups. This step has been realized by S. Franciosi, F. de Giovanni and L. A. Kurdachenko in [FdeGK 3]. In turn, the class of FC - groups is a subclass of the class of CC - groups. Therefore the study of just non-CC-groups, which has been initiated by L. A. Kurdachenko and J. Otal [KO 1, KO 2, KO 3], was a further natural prolongation of this research. This chapter is devoted to the main results of all mentioned above papers. As usual in order to avoid simple groups, we shall consider groups G with Fitt(G) * < 1 >. In this case, the first consequences of this assumption are collected in Chapter 10, with special mention to Theorem 10.5. The following lemma is crucial. 16.1. Lemma [KO 2] Let G be a group with a finite normal subgroup H such that GIH is a CC-group. Then G is a CC-group. In particular, a just non-CC-group has no non-identity finite normal subgroups. Proof
Let g e G. Put LIH = < gH >GIH . We want to show that GICG(gG) is a
165
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Chernikov group, so we only need to prove that GICG(L) is a Chernikov group. Since GIH is a CC-group, GICG{LIH) is a Chernikov group. Moreover, either LIH is a Chernikov subgroup or LIH includes a normal Chernikov subgroup TIH such that LIT is infinite cyclic [PY]. On the other hand, H is finite and so is GICG(H). Put K= Lr\CG(H).Then LIK is finite and KC\H in particular, GICG(KI(K n #)) is a Chernikov group. Thus so is G/C where C = CG{Kf)H) n CG(K/(KnH)). Since C stabilizes the chain < 1 > < Kf]H< K and ATD/^ is central in £, there exists the mapping 0 : C —>• Hom(K,KDH) given by 0(c) (JC) = [c,x], where c e C and x e AT with Ker9 = CG(AT) [KW, Chapter I, C]. Let « = \H\. Given C E C since [c,x] e / / for all x <E K, we have 1 = [c,x]" = [c,x"]. Thus A"" < KerQ and 0 induces the canonical homomorphism q> : C —• Hom(KIKn,KV\H) given by C\Z = < 1 > and < a,Z > = < a > xZ By Theorem 2.7 of the paper [FdeGK 2] < a,Z> includes a non-identity G-invariant subgroup M such that Mfl Z = < 1 >, contradicting Lemma 10.3. This implies that Z = £(//). Now let h e H\Z. By the above conclusion, there exists some y e H such that 1 * [h,y] = z\ e Z. Put [h,d] = z2 s Z. Since Z is locally cyclic, there exists u e Z such that < zi,Z2 > = < « > . We can find that < h > fl < w> = < 1>, so < h,u > = < h > x < u >. Therefore /!•>' = /i«' for some t =t 0. Similarly, /zrf = /JU*. Furthermore, ofyof"1 = yzj, with Z3 e Z, and then dy - yz->,d = z-^yd. It follows that }& = hz^d = A^. But /i* = (AM*)* = hu'+k, while A*d = Aw*"'. Since d inverts each element of Z and HIZ is torsion-free, this implies that t = 0, a final contradiction, which shows that H' = G and hence Z = C(ff)- This satisfies (i) and (ii). Let x e G. If 1 * v e £(G) and V = < v >; then K is an infinite cyclic subgroup and V is normal in G. Since G/F is a CC-group, [GIV,xV\ is a Chernikov subgroup and so [G,x] is minimax, that is G also satisfies (iii). Conversely, suppose that G is a group satisfying the conditions (i) - (iii). Since G is torsion-free and non-abelian, G is not a CC - group. Let H be a non-identity normal subgroup of G. Givenx e G\£(G) and put W = [G,x]. We note that Wis a minimax subgroup and W < C(G). Since G is a nilpotent group, M = H{M^{G) * < 1 >. Moreover MOW = < 1 >; otherwise < M,W> = MxW< f(G), contradicting (1). Therefore R = Mfl ^ * < 1 >. Now r0(W) = r0(R) so that WIR is a periodic factor-group of a minimax group. It follows that [G/R,xR] = [G,x]/W? = WIR is a Chernikov group. Since R < H, [GIH,xH] is a Chernikov group too and it readily follows that GIH is a CC-group. Since G is nilpotent of class 2, G/CG(X) S [G,X] for every x e G. It follows that G/Ca(x) is minimax by condition (iii). In other words, G Aos minimax conjugacy classes.
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16.3. Corollary [FdeGK 3] Let G be a group with a non-identity center. Then G is a just non-FC-group if and only if the following conditions hold: (i) ((G) is a torsion-free locally cyclic subgroup; (ii) GI((G) is a torsion-free abelian group; (Hi) for every x the subgroup [G,x] is cyclic. Proof Since every just non-FC-group is also a just non-CC-group, G satisfies (i) and (ii) by Theorem 16.2. Let x e G. If x e ((G), then [G,x] =< 1 >. Let x is an FC-group, the subgroup < x,z >G I < z > satisfies Max, so that < x >G has the same properties. In particular [G,x] is finitely generated and hence it is infinite cyclic. The proof of sufficiency is similar to the proof of sufficiency of Theorem 16.2. 16.4. Corollary [RZ] Let G be a group with non-identity center. Then G is a just non-(finite-by-abeliari) group if and only if the following conditions hold: (i) ((G) is a torsion-free locally cyclic subgroup; (it) GI((G) is a torsion-free abelian group; (Hi) [G, G] is infinite cyclic. Proof By Theorem 16.2 G satisfies (i) and (ii). Let l * z e [G,G]. Since Gl < z > is finite-by-abelian, [Gl < z >, Gl < z >] is finite. It follows that [G, G] is cyclic-by-finite. Since [G, G] is locally cyclic, [G, G] is infinite cyclic. Conversely, let G be a group satisfying conditions (i) - (iii), H a non-identity normal subgroup of G. Then Hn ((G) * < 1 >. Since [G,G] < ((G) and ((G) is locally cyclic, H D [G, G] * < 1 >. It follows that GIH is finite-by-abelian. These above results describe in a satisfactory way the case ((G) * < 1 >. In the sequel we study the more difficult complementary case in which ((G) = < 1 >. As we mentioned above in this case the usual condition Fitt(G) * < 1 > is assumed. 16.5. Theorem [KO 2] Let G be a just non-CC-group. IfFC(G) ((G) = < 1 >, then G is a just non-Chernikov group.
±but
Proof Let 1 * x e FC(G),X = < x >G . Then \G : CG(X)\ is finite and X is central-by-finite. By Schur's theorem (see, for example, [RD 9, Theorem 4.12]) [X,X] is finite. Lemma 16.1 yields that [X,X\ = < 1 >. In other words, X is a finitely generated abelian subgroup. The periodic part of X is a finite G-invariant subgroup, so Lemma 16.1 implies that Jf is torsion-free. Put C = CG(X). We can assume that X is Z-irreducible. Since GIX is a CC-group, all its elements of finite order form a subgroup TIX [PY]. Put Tx = T(\ C. Then [TX,T{[ is periodic (see, for example, [RD 9, Corollary to Theorem 4.12]). In particular, [TuTi] nX = < 1 >. Lemma 10.3 implies that [T\,T\] = < 1 >. The same arguments show that
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169
T\ is torsion-free abelian subgroup. Since T\IX is periodic, then T\ has finite 0 -rank. It is easy to see that CG(T\) = CGVO = C. Since the periodic part of a CC-group includes the derived subgroup [PY], the factor-group (C/X)/(Ti/X) = CITy is abelian. It follows that CIT\ < {(G/Ti), because CITi n[G/TuG/Ti] = < 1 >. Suppose that C * T\. Let LIT\ be a non-identity locally cyclic subgroup of CIT\. Since T\ < £(Z,), L is abelian. Since CIT\ < £(G/Ti), L is normal in G. Suppose that T * T\. By Theorem 2.7 of the paper [FdeGK 2] there exists a non-identity ^-invariant subgroup Q < L such that Q fl T\ = < 1 >, because TICT(L) is finite. The inclusions [L,T\ < Tx and [0,7] < Q imply [Q,T\ = < 1 >, in particular, CL(T) * < 1 >. By the choice of X we have CL(T)V\X= < 1 >. This contradicts Lemma 10.3, since the subgroup CiiT) is G - invariant. Thus T= Ti. In other words, T < C and GIC is abelian and torsion-free. Then GICGU-) is abelian. Put U = I ® z Q / = r ® z Q . The application of Theorem 1' of the paper [ZD 2] gives the decomposition U = V (B W where W is a Q - submodule such that G = CG(W)- Since U is an essential extension of L, WPi L * < 1 >. But this means that £(G) * < 1 >. This contradiction proves that C = T. For y e G\C put / / / I = C G/X (< y >° XIX). Let E = Hp\C and consider the mapping 6 : E —* E given by ed = [e,y], e € E. Clearly 6 is a Z / / endomorphism of E with ATerfl = Cf(y) and Imd = [E,y]. Since [E,y] < X, EICEiy) is a torsion-free finitely generated abelian group. If 6 is not injective and Ei = C^(y) we note that E\ *< 1 > and that £ i has only finitely many conjugates in G, say {Ef,... ,£Jf")-. Put Ej = Ef fl... fl£f"; then by Remak's theorem E/E2 <E/Ef
x... x £ / £ f .
Since £ / £ f = Egl/Ef = EIE\, ElEi is a torsion-free finitely generated abelian group. Since X and also C are Z-irreducible, £2 = < 1 >• It follows that E is finitely generated. If 9 is injective then E = [£,.y] < X, and so it is clear that E is finitely generated in this case. Since X * < 1 >, GIX is a CC-group, so GIH is a Cheraikov group and CIE is a Chernikov group too. All these facts imply that C is a minimax subgroup. Let U be a non-identity normal subgroup of G. By Lemma 10.3 t/ fl X * < 1 >. Since a periodic minimax group is a Chernikov group, CI(U(~\X) is a Chernikov group. It follows that GIU is a Chernikov group. In other words, G is a just non-Chernikov group. The above result naturally raises the question about the structure of just non-CC-groups with an identity FC-center. As in other cases of just non-A"-groups, the strategy will consist in splitting into two complementary cases:
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170
the non-monolithic case and the monolithic case. 16.6. Lemma [KO 1] Let G be a non-monolithic just non-CC-group, A = FittiG) * < 1 > and suppose that A is not torsion-free. Ifa\,... ,a„ e A,B = < a\ >G ... < a„ >G , then B is a just infinite ZH - module where H = GIA. Proof Let T be the periodic part of A. If FC(G) * < 1 >, then Lemma 10.3 yields that TTl FC(G) *• < 1 >. But in this case G includes a non-identity finite normal subgroup, which contradicts Lemma 16.1. Thus FC(G) = < 1 >. Corollary 10.6 imply that A is an elementary abelian p-subgroup for some prime p. Let M = {C | C is a non-identity G-invariant subgroup of B}. Since G is a non-monolithic group, M. * 0, and we can choose C e M.. Since GIC is a CC-group, < a, > G CIC is a Chernikov group [PY], so it is finite. It follows that BIC is finite. Since G is a non-monolithic group, (~\M = < 1 >, hence, 5 is a just infinite Z / / - module. 16.7. Lemma [KO 1] Let G be a non-monolithic just non-CC-group with FC{G) = < 1 >, A = Fitt{G) * < 1 >, ai,... ,a„ e A, B = < a\ >G ... < a„ >G ,H = GIA. Suppose that A is torsion-free. If the ZH module A is Z-irreducible, then B is a just infinite ZH - module. Proof
As above, put M = {C | C is a non-identity G-invariant subgroup of B}.
Again it is sufficient to prove that BIC is finite for any C e M. Since GIC is a CC-group, < a, > G CIC is a Chernikov subgroup because AIC is periodic. Since A is abelian, < a, >G CIC is a bounded group, so that < a, > G CIC is finite. Hence and BIC is finite. 16.8. Lemma [KO 1] Let G be a non-monolithic just non-CC-group with FC(G) = < 1 >, A = Fitt(G) * < 1 >, ai,... ,a„e^, 5 = < ai > G ... < a„ >G ,H = GIA. Suppose that A is torsion-free. Then B includes a G-invariant subgroup C * < 1 > such that BIC is Chernikov-by-polycyclic, and ZH - module C is just infinite. Proof Let £ be a non-identity G-invariant subgroup of B. Since GIE is CC-group then < a, > G EIE includes a G-invariant Chernikov subgroup EJE such that (< a, > G E/E)/Et is cyclic [PY]. Then BIE includes a G-invariant Chernikov subgroup FIE = (E\... E„)IE such that BIF is finitely generated,
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171
moreover ro(B/F) < n. We note also that ro(B/E) = ro(B/F). Let [/be a non-identity G-invariant subgroup of B with ro(B/U) is maximal. Since GIU is a CC-group, we can obtain that BIU again includes a G-invariant Chernikov subgroup VIU such that BIV is finitely generated and torsion-free. Let W be a non-identity G-invariant subgroup of V. By the choice of V it follows that r0(BIU) = rQ{BIW). In particular, n(UIW) = 0, in other words, £//Jf is periodic. This means that a Z//-module t/ is Z- irreducible. Let Y be a non-empty finite subset of U, C = < F > G .By Lemma 16.7 the ZH - module C is just infinite. By above BIC is Chernikov-by-polycyclic. Lemmas 16.6, 16.7, 16.8 show the role of just infinite modules in the study of just non-CC-group. Now we can use results of Chapters 6 and 7. 16.9. Lemma [KO 1] Let G be a non-monolithic just non-CC-group, A = Fitt{G) * < 1 >, and suppose that A is an elementary abelian p-subgroup for some prime p, H = GIA. Then Op(H) = < 1 >. Proof Suppose the contrary. Then there exists a normal subgroup P of G such that P * A and PI A is a finite p-group. By Corollary 10.6 A = Ca(A). Therefore there exists an element a e A such that P % CG{O). Thus if B - < a >G , then P $ CG(B). Since CG(B) > A, this gives that PCG(B)/CG(B) is a non-identity finite normalp - group. Hence OP(GICG(B)) * < 1 >. Lemma 16.6 yields that B is a just infinite ZH - module. Theorem 6.15 implies that in this case OP(GICG(B))
= < 1 >. We arrive to a contradiction.
16.10. Lemma [KO 1] Let G be a non-monolithic just non-CC-group, A = Fitt{G) * < 1 >,FC{G) = < 1 >,H = GIA. Suppose that A is an elementary abelian p-subgroup for some prime p. Then Opi(H) is finite. Proof Suppose that Opi{H) is infinite. Since G is non-monolithic, A includes a proper non-identity G-invariant subgroup B. Let a € A\B. Since GIB is a CC-group and aP = \,G BIB = CIB * < 1 > is finite. Let Q = CH(CIB), so that HIQ is finite. In particular, Q fl Opi(H) * < l >. It follows that Q f) Opi(H) includes a non-identity finite G-invariant subgroup L. Maschke's theorem (see, for example, [CUR 1, Theorem 10.8]) implies that C = Bx D for some L-invariant subgroup D. If x e L, then [D,x] < D and since x centralizes CIB,[C,x] < B. This gives [D,x] , so CA(L) * < 1 >. By Maschke's theorem we have the direct decomposition A = CA(L) XE where £ is a /.-invariant subgroup. It follows that E < [A,L]. Since L is normal in H, CA{L) and [A,L] are G-invariant subgroups of A. They intersect by identity, so that [A,L] = < 1 >. Corollary 10.6 implies that L < CG{A) = A. This is a contradiction.
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16.11. Corollary [KO 1] Let G be a non-monolithic just non-CC-group, A = Fitt{G) * < 1 >, FC{G) =,H= GIA. Suppose that A is an elementary abelian p-subgroup for some prime p.IfT is a periodic normal locally soluble subgroup ofH, then T is finite. In fact, by Lemma 16.9 Op(T) = < 1 >, and Lemma 16.10 proved that Opi(T) is finite. Lemma 14.13 implies the finiteness of T. 16.12. Corollary [KO 1] Let G be a non-monolithic just non-CC-group, A = Fitt{G) * < 1 >, FC{G) = < 1 >, H = GIA. Suppose that A is an elementary abelian p-subgroup for some prime p. Then H is an FC-group. Proof Given an x e H, we put X = < x >H . Then X is Chernikov-by-cyclic [PY]. If D is the divisible part of X, then D is a periodic divisible abelian normal subgroup of//. By Corollary 16.11 D has to be finite and so identity. Hence X is finite-by-cyclic and therefore H is an FC-group. Now is the turn of the case when Fitt{G) is torsion-free. 16.13. Lemma [KO 1] Let G be a non-monolithic just non-CC-group, A = Fitt(G) * < 1 >, FC(G) = < 1 >, H = GIA. Suppose that A is torsion-free abelian. If A includes a non-identity G-invariant subgroup B such that B is Z-irreducible as a ZH - module, then the ZH - module A is also Z-irreducible. Proof It clearly suffices to show that AIB is periodic. Suppose that it is false and choose an element aB of AIB such that \aB\ is infinite. Since GIB is a CC-group, EIB =< a >G BIB includes a G-invariant Chernikov subgroup UIB such that Ell] is cyclic [PY]. In particular, the ZH - module U is Z-irreducible. Since E/U is infinite cyclic, \G : CG(U/B)\ < 2 and UIB is Z-irreducible too. Since FC{G) = < 1 >, we have that G/CG(B) is infinite. Corollary 4.4 yields that E includes a G-invariant subgroup C * < 1 > such that UC\ C = < 1 >. But this contradicts Lemma 10.3. 16.14. Corollary [KO 1] Let G be a non-monolithic just non-CC-group, A = Fitt{G) * < 1 >, FC{G) = < 1 >,H = GIA , 1 * a e A,B = < a >G . Suppose that A is torsion-free abelian. Then B is a just infinite "LH - module and AIB is periodic. In particular, the ZH- module A is Z-irreducible. Proof Corollary 16.8 implies that B includes a G-invariant subgroup E such that BIE is Chemikov-by-polycyclic and E is a just infinite Z//-module. Let TIE be the periodic part of BIE; then the Z//-module T is Z-irreducible. By Lemma 16.13 the ZH- module A is Z-irreducible. Lemma 16.7 yields that B is a just infinite ZH-
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module. 16.15. Corollary [KO 1] Let Gbe a non-monolithic just non-CC-group, A = Fitt{G) * < 1 >, FC(G) = ,H=G/A,l±aeA,B=G.IfA torsion-free abelian, then CG(B) = A.
is
Proof Corollary 16.14 yields that AIB is periodic. Let b e A, g e CG(B), b\ = bg. There is a number n e N such that b" e B. We have now b\ = {b%y = (b")g = b". It follows that b\ = b since A is torsion-free. Hence CG(B) = CG(A) = A. 16.16. Corollary [KO 1] Let Gbe a non-monolithic just non-CC-group, A = Fitt{G) * < 1 >,FC(G) = < 1 >,H = GIA. If A is torsion-free abelian, then H is an FC - group. Proof Let 1 * a e A,B = < a >G . Corollary 16.14 shows that B is a just infinite ZH - module. By Corollary 16.15 CG(B) = A. Put M. = {E | E is a non-identity G-invariant subgroup of B}. Let x e H,X = < x >H . Since His a CC-group, we can observe that either Xis a Chernikov subgroup or X includes a normal Chernikov subgroup Y such thatX/7 is infinite cyclic [PY]. Let D be the divisible part of X. If E e M, then B/E is finite, so that \H : CH(BIE)\ is finite. It follows that D n CH(BIE) = D so D < CH{BIE) or [B,D] < E. Since it is valid for every E e M, then [B,D] < f] M = < 1 >. In other words, either X is finite or finite-by-cyclic. Consequently, H is an FC-group. 16.17. Corollary [KO 1] Let G be a non-monolithic just non-CC-group, A = Fitt{G) * < 1 >, FC{G) =,H = GIA. If A is torsion-free abelian, then H is central-by-finite and almost torsion-free. Proof Let 1 # a e A,B = < a >G . Corollary 16.14 shows that B is a just infinite ZH - module. By Corollary 16.15 CG(B) = A. Corollary 16.16 yields that H is an FC-group. Therefore it suffices to use Corollary 6.16. 16.18. Lemma [KO 1] Let G be a non-monolithic just non-CC-group, A = Fitt(G) * < 1 >,FC{G) = < 1 >, \ ± a a . Suppose that A is torsion-free abelian. IfA^zAe C,(GIA) and U = C G (< z >G BIB), then Uf\A is a just infinite ZG - module. Proof Firstly we note that A = CG(A) by Corollary 10.6. Since zA e C,(GIA), the mapping 6 : a —* [a,z], a e A, is a ZG - homomorphism, in particular,
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Kerd = CA{Z) and ImQ = [A,z] are G-invariant subgroups of A. It is easy to see that CA(Z) is a pure subgroup of A. Corollary 16.14 shows that CA(Z) = < 1 >, i.e. 9 is a monomorphism. In other words, A and [A,z] are G-isomorphic; similarly Uf)A =zo [Ur\A,z]. From the choice of z it follows that [Uf]A,z] < B. This means that [UC\A,z] is a just infinite ZG - module, hence and UnA is a just infinite ZG - module too. 16.19. Corollary [KO 1] Let G be a non-monolithic just non-CC-group, A = Fitt(G) * < 1 >,FC(G) = < 1 >. If A is torsion-free abelian, then A is just infinite ZG - module. Proof Corollary 10.6 gives the equality A = CG(A). Corollary 16.17 implies that GIA is central-by-finite and almost torsion-free. Let 1 ±aeA,B=G , A ± zA e C(GA4) and \zA\ is infinite, UIB = CG/B(< z >G BIB). Lemma 16.18 yields that Iff) A is a just infinite ZG-module. In particular, {UC\A)IB is finite. It follows that UIB is an FC - group, since U/(Uf]A) = UA/A < GIA and GIA is central-by-finite. Since UIB is not periodic, £(£///?) * < 1 >; moreover, (UIB)lt;(yiB) is periodic (see, for example, [RD 9, Theorem 4.32]). By Corollary 16.17 U/(Up\A) is central-by-finite and almost torsion-free, in particular, the periodic part 77(1/n A) of U/(Uf) A) is finite. Since (U n A)IB is finite, TIB is the periodic part of UIB. Let ZIB = {(UIB); then Z/B = (Z/B n 775) x X/B (see, for example, [FL 1, Theorem 27.5]). Let / = \ZIB n TIB\; then {ZIB)' = 175 < X/5, in particular, YIB is a non-identity torsion-free normal subgroup of GIB. Since the derived subgroup of CC-group is periodic [PY], YIB < C,{GIB). Let B±yB e YIB; then C G /B(VB) = GIB. By Lemma 16.18 ^ is a just infinite TLG module. 16.20. Lemma [FdeGK 3] Let G be a non-monolithic just non-FC-group, A = Fitt{G) * < 1 >, H = GIA, FC(G) = < 1 >. Suppose that A is an elementary abelian p-subgroup for some prime p. IfG is not periodic then A is a just infinite ¥PH- module. Proof We note that, by Corollary 10.6 H can not be periodic. In this case its center f (//) contains an element z of infinite order (see, for example,[RD 9, Theorem 4.32]). The mapping q>: a —• [a,z], a e A, is a G-endomorphism of A. Again Im
. From the G isomorphism [A,z] = 7m
, so that
= [C,z] are G-isomorphic. Since [C,z] < B, then the WPH - submodule [C,z] is just infinite. It follows that and C is a just infinite ¥PHsubmodule. Thus CIB is finite and then AIB is finite. Therefore A is a just infinite ¥PH - module. In the next results we are developing the basic features of the non-monolithic case. 16.21. Theorem [KO 1] Let G be a non-monolithic just non-CC-group with C(G) = < 1 > andFitt(G) * < 1 >. (1) IfFitt(G) = A is a non-torsion-free subgroup, then G is a just-non-FC-group. (2) IfFitt(G) = A is torsion-free then either G is a just non-FC-group or G is a just non-Chernikov group. Proof (1) Corollary 10.6 yields that A is an elementary abelian/? - subgroup for some prime p. By Corollary 16.12 GIA is an FC-group. Let U be a non-identity normal subgroup of G,V=ADU. By lemma 10.3, V*• < 1 >. Let x e G\V,XIV = < x >G VIV. Since GIV is a CC-group, XIV includes a normal Chernikov subgroup YIV such that XIY is cyclic [PY]. Since A is an elementary abelian then (XIV) fl (AIV) is finite. Since GIA is an FC-group, XAIA is finite or finite-by-cyclic. It follows that XIV is finite or finite-by-cyclic too. Hence GIV, and therefore G/U, is an FC-group. (2) Corollary 10.6 yields that ,4 is a torsion-free abelian subgroup. Suppose that FC(G) = < 1 >. By Corollary 16.17 H= GIA is central-by-finite and Corollary 16.19 implies that A is just infinite Z77 - module. Let again U be a non-identity normal subgroup of G, V = A fl U. By Lemma 10.3, V * < 1 >. Theny4/Kis finite and therefore GIV is a finite-by-FC-group, i.e. GIV is also FC-group. Hence GIU is also an FC - group. If G is an FC-hypercentral group, then G is a just-non-Chernikov group by Lemmal6.5 16.22. Corollary Let G be a non-monolithic just non-CC-group with f(G) = < 1 > and Fitt(G) = A * < 1 >. Suppose that A is an elementary abelian p subgroup for some prime p. IfG is locally soluble, then (1) A is a just infinite ¥PH - module where H = GIA ; (2) H is central-by-finite and almost torsion-free ; (3) every proper factor-group ofG has a finite derived subgroup. Proof By Corollary 10.6 A = CG(A). Suppose that H is periodic. Corollary 16.11 shows that in this case H is finite. It follows that A includes a non-identity finite G - invariant subgroup, what contradicts to Lemma 16.1. Hence H is not periodic. Lemma 16.20 implies that A is a just infinite ¥PH - module. Finally,
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Corollary 6.19 shows that H is central-by-finite and almost torsion-free. Let U be a non-identity normal subgroup of G, V = A f) U. By Lemma 10.3 V * < 1 >. Since A is just infinite then AIV is finite. It follows that GIV is finite-by-(central-by-finite). By Schur's theorem, a central-by-finite group has a finite derived subgroup (see, for example [RD 9, Theorem 4.12]). Therefore, GIV and hence G/U has a finite derived subgroup. 16.23. Corollary Let G be a non-monolithic just non-CC-group with C(G) = < 1 > and Fitt(G) = A * < 1 >. Suppose that G is not a just-non-Chernikov group and A is torsion-free. Then (1) A is a just infinite ZH- module where H = GIA ; (2) H is central-by-finite and almost torsion-free; (3) every proper factor-group ofG has a finite derived subgroup. Proof Again A = CG(A) is assured by Corollary 10.6. Since G is not just non-Chernikov, then FC{G) = < 1 >. Corollary 16.17 yields that H is central-by-finite and almost torsion-free, and Corollary 16.19 implies that A is a just infinite ZH - module. The proof of (3) is the same as in Corollary 16.22. 16.24. Theorem [RZ] Let G be a non-monolithic just non-(finite-by-abelian) group with £(G) = < 1 > and Fitt{G) * < 1 >. Then there is a subgroup X such thatXHA = < 1 > and \G : XA\ is finite. In other words, G nearly splits over A. Proof Clearly G is a just non-FC-group, therefore Corollaryl0.7 yields that either A is an elementary abelian p - subgroup for some prime p or A is torsion-free and abelian. In both these cases CG(A) -A. Since GIA has a finite derived subgroup, it is nilpotent-by-finite. Let A be elementary abelian. If we assume that GIA is periodic, then Corollary 16.11 yields that GIA is finite, what is impossible. Thus GIA has elements of infinite orders. By Lemma 16.2 A is just infinite ¥PH - module where H = GIA. Corollary 6.19 shows that H is central-by-finite and almost torsion-free. If A is torsion-free, then A is a just infinite ZH - module by Corollary 16.23 and also H is central-by-finite and almost torsion-free as well. Let 1 * zA £ £(G/A). The mapping , so cp is injective. Since Im
. Now |.¥[G,G]^ : XA\ is finite and G/[G,G]XA is finite, being a finitely generated abelian periodic group. Hence \G : XA\ is finite. Finally, XC\A = CA(Z) = < 1 >, so the proof is complete. 16.25. Corollary [RZ] Let G be a non-monolithic just non-(finite-by-abeliari) group with f (G) = < 1 > and Fitt(G) = A *• < 1 >. Then every proper factor-group ofG is central-by-finite. Proof. By Corollary 10.7 A = CG{A). As in Theorem 16.24 H = GIA is central-by-finite and A is a just infinite J,H - module. Let X be a subgroup such that/1 f l l = < 1 > and \G : XA\ is finite. This subgroup exists by Theorem 16.24. In particular, X i s central-by-finite. Put Z = C,{X). Let U be a non-identity normal subgroup of G, V = A f] U. By Lemma 10.3 V * < 1 >. Since A is a just infinite ZH - module, AlV is finite and A/VDXVIV = < 1 >. It follows that the index \AXIV : XVIV\ = \A/V : XVIV\ = \AIV\ is finite. Furthermore, the index \GIV : XVIV\ = \G :AX\ is finite. Since \X : 2\ is finite then GIV includes the abelian subgroup ZVIV of finite index. It follows that G/Vis central-by-finite (see, for example [TM 1, Lemma 7.5]). Therefore GIU'xs, also central-by-finite. 16.26. Corollary Let G be a non-monolithic just non-CC-group with C, (G) = < 1 > andFittiG) = A * < 1 >. (1) If A is periodic and G is locally soluble, then every proper factor-group of G is central-by-finite. (2) If A is not periodic, then either G is a just non-Chernikov group or every proper factor-group ofG is central-by-finite. 16.27. Corollary [FdeGK 3] Let G be a non-monolithic just-non-FC-group with Z(G) = < 1 > andFitt{G) = A * < 1 >. (1) If A is periodic and G is locally soluble, then every proper factor-group of G is central-by-finite. (2) If A is not periodic, then every proper factor-group of G is central-by-finite. 16.28. Theorem [RZ] (1) Let G be a non-monolithic just-non-(central-by-finite) group with f(G) = < 1 > andFitt{G) = A i= < 1 >. Then A is just infinite ZH- module where H = GIA is central-by-finite and almost torsion-free and CH(A) = < 1 >. Moreover, there exists an abelian torsion-free subgroup Zsuch that Zf]A =< 1 > and the index \G : AZ\ is finite. (2) Conversely, let H be a central-by-finite and almost torsion-free group, A a just infinite IJH - module such that CH(A) = < 1 >. Then every extension of A by
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H including the given module structure is a non-monolithic just non-(centralby-finite) group with the identity center and the Fitting subgroup A. Proof (1) As in Corollary 16.19 and Lemma 16.20 we can prove that A = Ca(A) is a just infinite ZH - module where H = G/A is central-by-finite and almost torsion-free. By Theorem 16.24 there exists a subgroup X such that XC\A = < 1 > and \G : AX\ is finite, in particular, X is central-by-finite and almost torsion-free. Therefore £(X) includes a torsion-free subgroup Z such that the index \X : Z\ is finite. Hence and index \G : AZ\ is also finite. (2) Let G be an extension of A by H, U a non-identity normal subgroup of G, V = AC\U. Since CG{A) = A, then V± < 1 >. The factor-group GIV is an extension of the finite subgroup AIV by the finite-by-abelian group G/A, hence G/Vhas a finite derived subgroup. Thus and G/U has a finite derived subgroup. Clearly, G is non-monolithic. If we assume that £(G) * < 1 > then £(G) C\ A •*• < 1 >. However it contradicts Corollary 6.5. Finally, by Corollary 10.7 the subgroup Fitt(G) is abelian and includes A, so that A = Fitt(G). Now we can use Corollary 16.25. In order to complete our study of just non-CC-groups, we only need to consider the monolithic case. To deal with it, we can use some results from Chapter 3. 16.29. Lemma [KO 3] Let G be a monolithic just non-CC-group with £(G) = < 1 > andFitt{G) * < 1 >. Then Fitt(G) is the monolith ofG. Proof Let M be the monolith of G; then M is abelian. Suppose that M * Fitt{G) = A.VxxXH = G/A, then H is a CC - group. We regard A as a ZH module, because A is abelian by Corollary 10.6. In this notations M is a simple 7LH - submodule of A. By Corollary 10.6 either A is an elementary abelian p-subgroup for some prime pox A is a torsion-free abelian subgroup. Suppose that A is an elementary abelian p-subgroup. Since GIM is a CC - group, AIM includes a non-identity G - invariant finite subgroup. In particular, AIM includes a minimal finite G - invariant subgroup B/M. By Corollary 4.3 there is a G - invariant non-identity subgroup C such that B = M x C, in particular, M f l C = < 1 >. But this contradicts Lemma 10.3. If A is torsion-free, it turns out that M is also divisible. In particular, A = M x D, for some subgroup D (see, for example, [FL 1, Theorem 21.2]). It follows that AIM is torsion-free and in this case AIM includes a non-identity G-invariant subgroup VIM which is infinite cyclic. By Corollary 4.4 U must include a non-identity G - invariant subgroup Fsuch that M R V = < 1 >, which again leads to a contradiction. Consequently, M = A. 16.30. Theorem [KO 3] Let G be a monolithic just non-CC-group with f(G) = < 1 > andFitt{G) = A * < 1 >. Suppose that A is not torsion-free. Then
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(i) there is a prime p such that A is an elementary abelian p-subgroup; (ii) A is the unique minimal normal subgroup ofG; (in) A = CG(4); (iv) if H = GIA and S = Socab{H), then S is a pi-subgroup including a subgroup R such that SIR is locally cyclic and Coren(R) = < 1 >; (v)ifFitt(H) * < 1 >, then G splits conjugately over A. Proof By Corollary 10.6 A is an elementary abelian p - subgroup for some prime p and CG(A) = A. Lemma 16.29 yields that A is the monolith of G. From Theorem 3.10 we obtain (iv). Finally, let LIA = Fitt(GIA). Suppose that £(L) * < 1 >. Then A < £(L) , so L < Fitt(G) = A. This contradiction shows that C,(L) = < 1 >. Theorem 4.5 proves now (v). 16.31. Theorem [KO 3] Let G be a monolithic just non-CC-group -with C, (G) = < 1 > andFitt(G) = A * < 1 >. Suppose that A is torsion-free. Then (i) A is a divisible torsion-free abelian subgroup; (ii) A is the unique minimal normal subgroup ofG; (Hi) A = CG(A); (iv) ifH = GIA andS = Socab(H), then S includes a subgroup R such that SIR is locally cyclic and CoreH(R) = < 1 >; (v) G splits conjugately over A. Proof By Corollary 10.6 A is torsion-free abelian and A = CQ(A). Lemma 16.29 yields that A is the monolith of G. In particular, A is a minimal normal subgroup of G, so that A is a divisible subgroup. From Theorem 3.10 we obtain (iv), and Theorem 4.5 gives (v). One more time, as an easy consequence of the above theorems, we obtain the following results. 16.32. Corollary [FdeGK 3] Let G be a monolithic just non-FC-group with C(G) = < 1 > andFitt(G) = A *• < 1 >. Suppose that G is locally soluble. Then (i) A is the unique minimal normal subgroup ofG; (ii)A = CG(A); (iii)G splits conjugately over A; (iv) if H = GIA, S = SocH, then S includes a subgroup R such that SIR is locally cyclic and CoreH(R) = < 1 >. Moreover, ifA is an elementary abelian p subgroup, then S is a p1-subgroup. 16.33. Corollary Let G be a monolithic just non-(finite-by-abelian) group with C, (G) = < 1 > andFitt(G) = A * < 1 >. Suppose that G is locally soluble. Then (i) A is the unique minimal normal subgroup ofG;
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(ii)A = CG(A); (Hi) G splits conjugately over A; (iv) if H = GIA, S = SocH, then S includes a subgroup R such that SIR is locally cyclic and Core H(R) = < 1 >. Moreover, ifA is an elementary abelianp subgroup for some prime p, then S is a pi- subgroup. If G satisfies the hypothesis of Corollary 16.33, then the derived subgroup KIA ofH= GIA is finite. As a consequence, CIA = CGIA(K/A) has finite index in GIA and CIA is nilpotent. Note that D. J. S.Robinson and Z. Zhang [RZ] used the reduction to the subgroup C in the consideration of the monolithic case. The following problem arises in connection with the results of this chapter and Chapter 15 Question 6 < 1 >.
Describe the structure of a just non-PC-group G with Fitt(G) *
Chapter 17 Groups whose Proper Factor-Groups Have a Transitive Normality Relation
It is well known, that the relation "to be a normal subgroup" is not transitive. Therefore it is naturally to consider the groups, in which this relation is transitive. A group G is said to be a T- group, if it satisfies the following condition: (7) IfH is a normal subgroup ofG and K is a normal subgroup ofH, then K is normal subgroup ofG. In other words, a group G is a T - group if and only if every its subnormal subgroup is normal. The theory of T - groups goes back to the paper of R. Dedekind [DE 4] (clearly, a group, every subgroup of which is normal is a natural example of a T group) and has been continued in papers of E. Best and O. Taussky [BT], G. Zacher [ZG ], W. Gaschutz [GW], I. N. Abramovsky and M. I. Kargapolov [AK ], I. N. Abramovsky [A I ]. However the real progress in this area has been achieved by D. J. S. Robinson in his papers [RD 1, RD 5].Since different generalizations of r-groups have often arisen in many researches related to normality, their investigation still be very actual Continuing these researches, D. J. S. Robinson considered the soluble groups, every proper factor-groups of which is a T - group [RD 11]. Chronologically it was the third of the already investigated types of just non-^-groups. Soluble just non-r-groups play an important role in investigations related to study of the normal structure of groups and the class of such groups has been described as best as possible. In [RD 11] D. J. S. Robinson provided all details of its structure. Definitely this description requires very detail analysis and significant amount of work.. This class of groups has some peculiar properties. If just non-abelian groups were monolithic, and just infinite groups were non-monolithic, then both these cases meet combined already in the study of just
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just non-r-groups. Further, soluble T - groups are metabelian. The description of simple modules over metabelian groups in the general case is a complicated problem. Therefore it is interesting to obtain their description over some specific types of metabelian groups. The present chapter is devoted to presentation of results abut just non-r-groups from the paper [RD 11]. Since there are no too many classes of groups described so neat, we decided to keep all, even small details of their structure. For this, we will permanently apply the information on the structure of soluble T-groups given in [RD 1]. The main results about soluble T - groups will be given now. All details one can find in [RD 1]. An automorphism
maps each element to a power of itself. Let G be a soluble T- group; then it is metabelian [RD 1, Theorem 2.3.1]. Also L = [[G,G],G] is the last term of the lower central series of G, and GIL is a Dedekind group (that is the group, every subgroup of which is normal). Furthermore, CG(L) = CG([G,G]) = Fitt{G). The following three classes naturally appear in the case of non-abelian soluble T- groups (see again [RD 1]): (A) G is a periodic group; (B) C = CG([G,G]) is non-periodic (G is a non-periodic group of type I); (C) C = CG([G, G]) is periodic (G is a non-periodic group of type II). If G is a periodic soluble T - group, then L and GIL does not contain elements with the same odd prime order [ RD 1, Theorem 4.2.2] and the Sylow 2 subgroup of L is divisible. If G is a soluble T- group of type I, then C is abelian and G =< t,C >, where \G : q = 2, c' = c~l for each c e C, < t2,C2 > = < t2,C4 > [RD 1, Theorem 3.1.1]. If G is a soluble T - group of type II, then its structure is less known. However, C is abelian, [G,G] is divisible and C=[G,G]xB where B < ((G). If p e I1([G,G]) and Bp is a Sylow p - subgroup of B, then Bpp = < 1 >. If x e G, Cp is a Sylow/? - subgroup of C, c e Cp, then cx = ca, where a is an invertible/? - adic integer such that a = l(mod p"^); here ca is understood to mean c"[ where ai is an integer such that ai = a(mod \c\) [RD 1, Theorem 4.3.1]. Note also, that some other important facts about T - groups and their generalizations can be found in the survey of D. J. S. Robinson [RD 24]. We need the following technical lemma. 17.1. Lemma [RD 11]
Let G be group, H an abelian normal subgroup of G
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such that every subgroup ofCo(H) is normal in G. Assume also that for every prime p the Sylow p - subgroup ofH is either elementary abelian or not bounded. Then G is not a just non-T-group. Proof Suppose that G is a just non-r-group and let R be a non-normal subnormal subgroup of G. Then R $ Ca(H), so there exists b e R such that [b,H] * l.The element b induces a non-identity power automorphism in H, because H < CG(H). If H is not periodic, then from above hb = h~l for all h e H, and [H,,b\ = H2'. If s is the subnormal defect of R in G, then R > H1' * < 1 >, which implies that R is normal in G. Thus H is periodic and there is a prime p e Tl(H) such that a Sylow p - subgroup P of H is not centralized by b. There is a p - adic integer a such that hb = h" for each h e P [RD 1, Lemma 4.1.2]. With s as before, R > pCo-i)^ and consequently pC"-')1 = < 1 >. However, it implies that either P is elementary abelian and a = l(mod p) or P is not bounded and a = 1; in this case [P,6] = < 1 >. 17.2. Lemma [RD 11] Let G be a just non-T-group, M and U its normal subgroups. If one of the following conditions holds, then either MorU is identity. (f) M and U are periodic andU(M) n IT([/) = 0. (if) M is periodic, U is torsion-free and G/U is periodic. Proof Let H be a non-normal subnormal subgroup of G and suppose that M * < 1 >. In both cases (i) and (ii) MC\ U = < 1 >; thus (Hn M)U is normal in G. It follows that [77n M,G] < HC\ M and so Hf)M= < 1 >. Similarly Hf]U = < 1 >. If (i) is valid, then
//n(Mx LO = (/fnJW)x(i/nt/) =< 1 >, which gives H = (HM) n (HU). It follows that H is normal in G. If (ii) is valid, then H is periodic, because H = HUIU; therefore HM is periodic and (HM) C\U= , which implies that (HM) n (#£/) = # and again H is normal in G. • The study of soluble just non-r-groups is divided into some partial cases. The first case is about nilpotent just non-r-groups. 17.3. Theorem [RD 11] Let G be a nilpotent just non-T-group. Then G is a just non-abelian group with the exception of the quaternion group of order 8. Proof Suppose that G is not just non-abelian. Every nilpotent T - group is a Dedekind group, in particular, it has a finite derived subgroup (see, for example, [RD 19, 5.3.7]). Thus for this case we will use Corollary 16.4. By this Corollary G
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is torsion-free, [G, G] < £(G), £(G) is locally cyclic, and [G,G] = < d > is cyclic. Let/? be an odd prime; then the factor-group Gl < dP > is non-abelian and non-Dedekind, a contradiction. This contradiction shows that G is just non-abelian. Naturally, we must exclude the quaternion group of order 8, because it is a T - group. The next step is the consideration of soluble just non-r-groups G with no minimal normal subgroups. Denote by Z(p°°) the ring of integer p - adic numbers and by Q(px) the field of p - adic numbers. 17.4. Theorem [RD 11] Let G be a soluble just non-T-group. Suppose that G is non-nilpotent and does not include minimal normal subgroups. Then Q(px) includes a subfleld F and U(Z(pco)) includes a subgroup Y, satisfying the following conditions: (i) 7 * < -1 >; (ii) ifY+ is the additive group generated by Y, then Y+ * Q + Y+ = F; (Hi) the group G is isomorphic with the natural semidirect product ofF (as an additive group) by Y. Proof Let L= [[G,G],G], then L *< 1 > and GIL is a Dedekind group. Furthermore, the lower central series of G is stabilized on L. If H is a non-identity normal subgroup of G, then GIH is a soluble T - group and hence is metabelian. Thus G" < H, and either G" = < 1 > or G" is the monolith of G. The latter is impossible. Consequently, [G, G] is abelian, and hence L is abelian too. Suppose that L is not torsion-free, let p e 11(1), P = QAi(Z,).Then P is a non-identity G - invariant subgroup of L. Since G does not include minimal normal subgroups, P must have a descending series of G - invariant subgroups P = Pi> P2> ... Pa > Pa+\ > ... Py = < 1 >, where y is a limit ordinal. Let g & G, since GIPa is a T- group, g induces a power automorphism on PIPa Let a e P\Pi, then (aPi)s = akPi for some 0 < J f c < p - l . Clearly if a > 2, then (aPa)g = akPa. It follows that CG(aPa) = CG(aP2) for all a > 2, and hence CG(PIPt) = CG(P). This means that GICG(P) is finite, that is P < FC(G). But in this case G includes minimal normal subgroups, what contradicts our hypothesis. Thus L is torsion-free. Suppose that L is not Z - irreducible. Then L has a descending series of G - invariant subgroups
Groups whose Proper Factor-Groups Have a Transitive Normality Relation
185
L = L\ > Ll > ... La > La+l > ... Ly - < 1 >, where y is a limit ordinal. If a > 2, then [G/La,G/La] = [G,G]/La < L/La, so GILa is a soluble T - group of type I. Put C = CG(L), then C < CG(L/La). Note that for every T - group g the equality CQ([Q,Q]) = CQ([[Q,Q],Q]) is valid [RD 1, Lemma 2.2.2]. From this we may deduce that C < CG([G,G]ILa). Since G is non-nilpotent, G * C. Choose g e CAC. Then g 2. It follows that C is abelian and cg = c _1 for all c e C,g e G\C. In particular, every subgroup of C is G - invariant, and Lemma 17.1 leads us to a contradiction, which proves that L is Z - irreducible. Suppose that LP * L for some prime p. Then p ±2, because Z,/L2 is divisible [RD 1, Lemma 2.4.1]. Since GIL is nilpotent, all elements of finite order of GILP form a subgroup. If GIL is not periodic, GIU is a soluble T-group of type II [RD 1, Corollary 2 of Theorem 3.1.1]. But for this case LIIf is divisible [RD 1, Theorem 4.3.1]. This contradiction shows that GIL is periodic. Put Lm = n « e z ^ " ' ^ e n L/Lm is torsion-free, the Z- irreducibility of L implies La = < 1 >• If x e D = CoiLIW), then x induces an automorphism on each LIIP" whose order is a power of p. Since GIL is periodic, we conclude that x induces an automorphism on L whose order is a power of/?, that is DIC is a p group. However GIW is a periodic soluble T - group, therefore p £ Tl(GIL) [RD 1, Theorem 4.2.2]. It follows that D = C, in particular, G/C is a cyclic group and \GIC\ divides (p - 1). Let 1 ± a e L, A = < a >G , then ,4 is finitely generated. Suppose that H{GIL) contains an odd prime q. Then GIAq is periodic, q e Il(G/L) r\U(L/Ai). This contradicts Theorem 4.2.2 from [RD 1]. Thus Tl(G/L) = {2}. Let g e CAC. Then g induces in A/A3" a power automorphism whose order is a power of 2 and divides c"1, the latter being the only power automorphism of order 2. Since c c l r D « E N ^ 3 = < 1 >> * = ~ f° every c e A, unless [^4,g] = < 1 >. Since LIA is s 1 periodic, it follows that c = c" for all c e L,g e CAC. Next Z, = L2, thus LIA2 includes a Prufer 2 - subgroup. Let PA42" be a Sylow 2' subgroup of LIA2"', n e N; then P * L and L/F is an abelian divisible 2 - group. In this case CIA2" is nilpotent, so it is a Dedekind group. But CIA2'' also includes a Prufer 2 - subgroup. This means that CIA2" is abelian, and therefore C is also abelian. We have CIA2" = PI A2" x El A2" where El A2" is a 2-group. If g e CAC, then from the structure of a soluble T-2-group [RD 1, Lemma 4.2.1] we obtain g2P e C(G/P), and hence \g2,E] < A2". Since c« = c"1 for all c e Z,,[g2,L] = < 1 >. It follows that [g2,C]. If Cf] F = < 1 >,then [[G,G],F] < CD F = < 1 >, which is impossible. Thus C n V * < 1 >, and without loss of generality we can assume that V < C. Now G/C, and hence GIV, is non-periodic and GIM is non-abelian because M < [G,G]. Hence G/Vis also a soluble T - group of type II. It follows that CIV is abelian, and Remak's theorem implies that and C is abelian. Let g € G; then (aM)g = (aa*)M, a e [G,G], because [G,G] is a divisible abelianp subgroup. It follows that (cM)g = (c"*)M for c e C. But a» = aa« for all a e [G,G], therefore it is valid for [G,G]V/V and CIV. Hence c« = c"« for all c e C. Since C = CG([G,G]) and [G,G] is not bounded, Lemma 17.1 implies a contradiction. This means that G is a monolithic group with the monolith M. In turn, it follows that [G, G] is a Prufer/7 - subgroup. Put Z = C(C); then [G, G] < Z, and (2)
Z=[G,G]xZ)i
for some subgroup D\ (see, for example, [RD 19, 4.1.3]. Clearly D\M< C< G, so DiMis normal in G and [DUG] < [G,G] C\(DiM), which show that (3)
[DUG]<M
If d e £>i,g G G, then 1 = [rf.gp = \dP,g\, therefore D?x < C(G), in particular, D^ is normal in G. Since Mis the monolith of G, D^ = < 1 >, i.e. £>i is an elementary abelian p - subgroup. Let T/[G,G] be the periodic part of GI[G,G]; then C < T. Moreover, C * [G,G] by Lemma 17.1. The structure of a soluble T - group of type II provides the following information: {CI[G,G])P' = < 1 > for some e e N and ag = \{modpe) for all g e G [RD 1, Theorem 4.3.1]. Observe that e > 0, so that (4)
ag= \[modp).
Groups whose Proper Factor-Groups Have a Transitive Normality Relation
191
Now suppose that g s T . Since ag is an invertible p - adic integer satisfying (4), there are the following possibilities: either/) is odd and ag = 1 orp = 2 and ag e {1,-1}. Thus either T = C or |77q = 2, and we can write (5)
T =
where either r = 1 or a, = - 1 ; in either case t2 e C and |*2M| =1,2. Therefore *2 has order dividing 4. Let t * 1; if g e G, then / g = to where a e [G,G]. Hence ( is a normal subgroup of G. If f2 * 1, then M< < t2 >. Let p = 2 and [G,G] = < a„ \ 2 2 2 a\ = l,a „+l = a„,n e N >; then M= < a\ >. Clearly t e Z. If f e [G.G], then /2 = 1 or t2 = a\ because a'2 = a^1 if f * 1. Suppose t2 £ [G,G]. Since |/2| = 2or 4, it belongs to < cti > xD\. Since a\ 6 < i1 >, we can assume that f2 = a2U for some 1 * u e D\. Thus the possibilities for t2 are only \,a\,a2U. Since 77[G,G] is bounded, (6)
G/[G,G] = (TI[G,G\) x (YI[G,G\)
for some subgroup Y (see, for example, [RD 19, 4.3.9 ]. From (6) and (5) we obtain (7)
G=
Consider the case when C is abelian. Here C = Z = [G,G] x D\, and together with (7) becomes G = < t,D\,Y>. Put W = < t,Y>. Since D\ is an elementary abelian, D\ = (< t,[G,G] > C\Di) x D for some subgroup Z). Hence G = = WD. From the inclusions [G,G] < Y< W we obtain that W is normal in G. Since TC[Y= [G,G], we have Wp\D < < t,[G,G] > |~l D = < 1 >. Hence G = WD and W D £> = < 1 >. Now we consider the structure of W. First (8)
W/(WnC)= WCIC = GIC.
The map gC —• ag is an isomorphism of GIC with a non-periodic group T of p - adic integers all of which are congruent to 1 modulo/?. Now WPiC= Wf)([G,G]xDi)
= [G,G]x(fFn£>i).
Also WnDt
=< t,Y> nDi =< *,[G,G] > (iDi = < n£>i.
If/2 € [G,G], then Wfl-Di = < 1 > by the last equation; otherwise h = aw and < t2,[G,G]> DDi = < w >, so ^ n ^ i = < « >. Hence ^ f l C = [G,G]
Just non-X-Groups
192
or WHC = [G,G] x < u > according to whether t2 e [G,G] or t2 g [G,G]. Also, W/(Wf] C) = r by (8). Suppose that t2 = « i ; if there is an element rf e £> such that td * and ^ is a power automorphism of P. By Lemma 5.2.2 of [RD 1] G is a P - group. This contradiction proves that Q2CP) = Q\(P), in other words, P is an elementary abelian p - subgroup. By our hypothesis, [G,G] = CG(M), in particular, GICaiM) is abelian. By Theorem 2.3 G/CG(M) is a locally cyclic pi- group, in particular, it is countable. Since elements of G induce power automorphisms in the elementary abelian p subgroup PIM, CG{M)/(CG(M) D CG{PIM)) is cyclic of order dividing p - 1. If g e CG(M) n CG(P/M), then g induces in P an automorphism of order 1 or p. The equation P = CG(P) implies g e P. Thus (12)
CC(W) n CG[PIM) = CGGP) = P.
It follows that GIP is countable. Then there is a family {G„/P | « e N} of finite subgroups G„ satisfying the following conditions: if n < k, then G„IP < Gt/P and GIP = Q„ e N G„/P. Since G„/F is finite, there is a finite subgroup F„ such that G„ = F„P. Moreover, we can choose these subgroups such that F„ < Ft for n < k. It follows that F = |J„ e N F„ is a subgroup and G = PF. Clearly, P C\ F„ is the normal Sylow p - subgroup of F„, therefore F„ splits conjugately over {P fl F„) : F„ = (P fl F„)X„ by Schur - Zassenhaus theorem ( see, for example, [SM , Theorem 8.10]). Let k > n, then by Dedekind modular law F„ = (F„nP)(F„r\Xk). In particular, the subgroups (F„nXk) and X„ are conjugate in F„. In other words, we can choose the subgroups X„ such that X„ < Xk for n < k. Put X = |J„eN^"- T h e n F = (PnF) \ X and hence also G = P\X. Furthermore, CX(P)"= < 1 > by (12). Consider the situation when M is the monolith of G. Assume that P * M and let a e PXM. Since GIM is a P - group, AIM = < a > M is normal in G. If M is finite, then G/CG(A) is finite pi- group. Using Maschke's theorem (see, for example, [RD 19, 8.1.2]) we obtain the decomposition A = MxB where B is a G - invariant subgroup, which is impossible. If Mis infinite, the using Theorem 4.1 we obtain again that A = Mx B for some G - invariant subgroup B, These contradictions show that P = M. Thus G is a group of type (I). Consider next the opposite situation: let G include a non-identity normal subgroup U such that Mf] U = < 1 >. If Uf] P = < 1 >, then U < CG(P) = P; therefore Uf]P* < 1 > and we can assume that U < P. Also U =G UMIM shows that every subgroup of U is G - invariant, thus we can assume that \U\ = p. Also and M =G MUIU, so that |A4| = p. Suppose now that P * MU and choose a G P\MU. Since G/Mf/ is a T - group, ^1 = < a > MU is normal in G. Since ,41
196
Just non-X-Groups
is finite, GICG{Ai) is a finite/?'- group. Using again Maschke's theorem (see, for example, [RD 19, 8.1.2]) we obtain the decomposition^ = MUx B\ where B\ is a G - invariant subgroup. Then B\MIM = GB\UIU. Let g e G, then g induces in PIM and PIU power automorphisms which both have the form c —* c", c s P, because they must agree on 5 i M M a n d B\ UIU. Hence cg = c" for all c e P; this situation is impossible by Lemma 5.2.2 of [RD 1]. Hence P = Mx U and X is isomorphic with a subgroup of GLiip) what is diagonal because X induces a power automorphism groups in M and U; this subgroup is not scalar since it does not include a group of power automorphisms in P. Clearly p is odd and G is a group of type (II). Subcase (A II): PIM is abelian and P is nilpotent of class 2. Since PIM is abelian, [P,P] = M. If g € CG(P/M), then [\g,P],P] = < 1 > by (10), therefore by Three Subgroups Lemma ( see, for example, [RD 19, 5.1.10] [g, [P,P]] = < 1 >, that is [g,M] = < 1 >. It follows that for every a e P there is an element b e M such that cfi = ab, and also a«p = ah? = a. In other words, CG{PIM)/CG{P) is a p group. But (11) yields CG(P) < P\ therefore CG(PIM) < P and (13)
CG(PIM) = P.
Next, if p = 2, a periodic group of power automorphisms of PIM has order a power of 2 and equation (13) gives P = G, i.e. G is nilpotent. Thus p is an odd prime and therefore P is a regular p - group. If we assume that G includes a normal non-identity subgroup U such that Mf] U = < 1 >, then both groups PIM and PUIU are Dedekind group. Since they are also 2'-groups, PIM and PUIU are abelian. By Remak's theorem P is abelian. This contradiction shows that M i s the monolith of G. Since P * G, we can choose an element g e G\P, then g g CG{PIM) by (13). For every element aM e P/Mvte have (aM) g = (aM) a where 1 * a e [/(Z^ 0 0 )). If a,Z> e P, then (aM)s = {aa)M,{bM)z = (ba)M; hence (14)
[a,b]g = [a a >6°] = [«,6]" 2 ,
because [ ^ M ] = < 1 >. This means that the subgroup < [a,b] > is normal in G. Since M is the monolith of G, it follows that \M\ = p. Let M = < a >. Suppose that Pp * < 1 >; then M < Pp and consequently a = bp for some b e P, because P is regular. Now & = 6"c for some c e M. In turn a* = (A*y = (6 a )^ = a". But (14) implies ag = a" ; therefore a 2 = a(modp) and a = l(modp). If we assume that [PIM)pe = < 1 > for some e e N, the congruence ap'~ = l ( m o d p e ) implies thatg induces in PIM an automorphism of order of power ofp; therefore by (13) g e P. H PIM is not bounded, a must have finite order; together with a = l(mod/>). and p > 2 this implies a = 1 (see, for example, [FL 2, Theorem 127.5]). These
Groups whose Proper Factor-Groups Have a Transitive Normality Relation
197
arguments indicate that (15)
PP=.
Put Z = C,(P), clearly M < Z. Let 1 * a\ e Z; then < a\,M > is normal in G, in particular, it is finite. Using again Maschke's theorem, we come to contradiction. This contradiction proves the equation Z = M. Now PIM is elementary abelian by (15); hence P is an extra-special p - group. Choose a basis {xxM\ X e A} for PIM. Then (16)
[xx,x„] = a**">
where/is a non-degenerate alternating bilinear form. Since P = CQ(PIM), GIP is cyclic with order q dividing p - 1. Hence there is an element g such that |g| = q,G = < g > P and < g > C\P = < I >• Moreover, C(P) = < 1 >. The element g induces a power automorphism in the elementary abelian group PIM of the fornix —• x",x e P, where 1 < n < p. Thus (17)
{xxY = {xx)"a"\X e A
for certain integer rtx satisfying 0 < nx < p. Suppose that m * 0. Since / is non-degenerate, there is /i e A such that/^.p) * 0(modp). We will replace xi by a suitable element of the form x\ = (xxYix^y. It follows from (16) and (17) that {xx)s = (x^ya" where u = snx + trip + st{ n2 )_/(A,^). We want to choose s and t such that/? / 5 and/? J t but u = 0(mod/?). Consider the congruence (18)
xnx +ynn +z = O(modp)
where z = ( " )/(A,/i); notice that p K z. Since /» / rt*, we need only look for >> such that/? / (yrif, +z). If «^ = 0, anyy * 0 will do; if n^ •*• 0, we can choosey such that 1 - group, then/? = 2 [RD 1, Lemma 4.2.1]. Thus P is a 2 - group. In this case L/M is a divisible abelian 2 - group. Define C = CG([G,G]/M), then C/M is nilpotent of class < 2. Since the periodic part of C/(Z(2CC)) has order 2 (see, for example, [FL 2, Theorem 127.5), then \GIC\ < 2. Let x e CO i>. By (9) the mapping ipx:a —• [a,x], a e L, is a homomorphism such that Irwpx < M and .Kerpx > M. Since L/M is divisible and M is elementary abelian, the isomorphism Im.
The inclusion L < [G, G] < C n P proves that L is abelian. Since \G/C\ < 2, C contains all elements of G with odd order. Now CIM is a Dedekind group, therefore its Sylow 2'- subgroup QIM is abelian. Assume that Q * M. If [M, Q] = < 1 >, Q is nilpotent. Since 2 is normal in G, Lemma 17.2 proves that Q is a 2 subgroup, so Q = M Hence [M, Q] * < 1 >. Suppose that Z * M, then L/M is a non-identity divisible abelian 2 - group. In this case L2 * < 1 >, therefore LIL2 is divisible, which proves that L is divisible too. For every element x e C consider again the mapping , in particular, [M,Q] = < 1 >, a contradiction. It follows that L = M and GIM is a Dedekind group, thus C = CG([G,G]/M) = G and (19) becomes [L,P] = < 1 >. Since [M,Q] * < 1 >, Theorem 4.5 implies that G splits over M : G = MX and Mf)X= < 1 >. Therefore P = P n (MX) = M ( P n X ) - Since P is non-abelian, F f l l * < 1 >. Also [M,PnX] < [L,P] = < 1 >, so PHX is normal in M r = G. Thus G / ( P n ^ 0 is a T - group and the isomorphism M = G M(P f] X)I{P D X) shows that |Af| = 2. Consequently, M < C(C?). In other words, G is nilpotent. This contradiction proves that G is a 2 - group. Equation (19) yields (20)
[L, C] = < 1 > and C = CG{L).
If we assume that GIM is a Dedekind group, then L = M and C = G, (20) implies that G is nilpotent. Also LIM is divisible whenever L * M and this, as we have already proved above, implies that L is divisible. The factor-group GIM is a soluble T - group, furthermore GIM is a non-nilpotent 2 - group. By Lemma 4.2.1 of [RD 1] G = < C,t > where f2 e C,(cM)' = c~'M,c e C,C/Mis abelian and not bounded. Together with (20) it implies that C is nilpotent of class at most 2. Define a : a —• a" 1 ,a e [ , t : a —>• as,a e L. Then r - 1 cr is identity on L/M and T_1CT - 1 e Hom(L,M) = < 0 >. Therefore r =CTand (21)
ar = a-\aeL>
which proves that f e CG (•&/)• This permits us to conclude that M < £(G). Suppose there exists a non-identity normal subgroup U such that M f] U = < 1 >. Since U =G UMIM, we can assume that \V\ = 2. Also L $ U, so GIU is not a Dedekind group and its structure is similar to that of GIM. In particular, CU/U is abelian. Since it is valid also for CIM, Remak's theorem proves that C is abelian. It follows that a' = a~x for all c e C , which is impossible by Lemma 17.1. This contradiction shows that G is a monolithic group with monolith M, in
Groups whose Proper Factor-Groups Have a Transitive Normality Relation
199
particular, L is a Priifer 2 - group and \M\ = 2. Put now Z\ = f (C). Then L < Z\ and Zi = Lx D (see, for example, [RD 19, 4.1.3]). Suppose that £> contains an element d of order 4; then d' = d~la for some ae M. Hence (rf2)' = {drxa)2 = d~2 = d2. Since d2 e Zu it follows that d2 e £(G), in particular, < t > is a normal non-identity subgroup of G. But this is impossible, because < d2 > f W = < 1 >. In other words, D is elementary abelian. This implies that DMIM < C(G/M), so that (22)
[D,G]<M.
If d e D, the mapping p] =< 1 >= [M,G]. Moreover, lm. Since Hom{GIC,M) has order 2, we must have |£>| < 2. Put D = . Let now L = < a„ | a2 = ljfl^+i = a«« e N >. If c e C, then c' = c _1 a for some a e M. Hence c'2 = (c~la)~la = c. Since G = < t,C >, it follows that t2 e C(G); in particular, the subgroup < t2 > is normal in G. Therefore, either t2 = I or M < < t2 >. Also /2 s £(G) and since x < and the possibilities for t2 are \,a\ or a2<s? (if d * 1); indeed if cf * 1, then cf = aic/by (22) since < d > cannot be normal in G. Subcase (AII(a)): PIM is abelian, P is nilpotent of class 2 and C is abelian. Here C = Z\ = LxD, and d* 1 by Lemma 17.1. Since d' = a\d, we have (to?)2 = (f2)ai(c?2) = t2a\. Hence t2 = a\ implies (td)2 = 1.Therefore we can assume that either t2 = \ort2 = azd; \d\ = 2 or \a] = 8. Thus G is of type (IV). Subcase (A II(fi)): P/M is abelian, P is nilpotent of class 2 and C is nilpotent of class 2. Since CIM is abelian, [C,C] = M < Z\. If x,.y e C, then 1 = [x,y]2 = [x2,y]. This means that CIZ\ is elementary abelian. Choose a basis {xxZ\ | A e A}. Then (XA)2 = a^' where a e Z, and /' e {0,1}. Now a = b2 for some J e i and {xxb~1)2 = x^a -1 = d'. W r i t e r = xj.6 -1 ; then y\ = ^^' c f° r s o m e c e ^ a n d (y^)' = (y~j}c)2 = y'2. It follows that < d' > is normal in G, which can only mean that d' = 1 and y\ = 1. In short, we can assume that (23)
x\ = 1
for all X e A. Define X = < xA | A e A >. By (23) X2 = [X,X\; also C = XZU so A/= [C,q = [A;A] and M= [X,X] = X2. Suppose that ueXOZi and write M = x"k\... x\rra where a e M, «, e N, A, e A, 1 < / < r. The independence of xx,Z\ indicates that each w, is even; thus u e X2M = M. Consequently
200
(24)
Just non-X-Groups
Xf)Zi=M.
Therefore £(X) <Xn f(C) = Xfl Z, = M, and £(JQ = M Also, XIM = CIZ is an elementary abelian 2 - group. In other words, X is an extraspecial 2 - group generated by the elements of order 2. Clearly C is a direct product of X and Z\ in which Q(X) and < ai > are amalgamated. As we have already observed t2 = \,ort2 = a\, or t1 = ajd (if d * 1); in fact the second possibility can be discarded if t is chosen suitable. The argument for this has already been given in the last part of Theorem 17.7. Consider the mapping a : xM —• [x,t], x e X. Since t e CG(X/M) f] CG(M), a e Hom{XIM,M). lfd=p 1, one can assume that a = 0 and [X,t] = < 1 >. For this case if x\ = x\a\, we obtain (xxd)' = xxd while (xxd)2 = 1. Thus G is of type (V). In conclusion, observe that even if d * 1 one can still take a = 0 at the expense of losing x\ = 1; for {xiai)' = x^a2 if*l * *ANote that all types of the groups, obtained above in the Theorems 17.3 - 17.9, are just non-T-groups. But we will omit here the proof of this fact. In connection with the obtained above results the question on a structure of simple modules over soluble T- groups becomes actual. 17.10. Proposition [RD 11] Let X be a soluble T-group, F = Fitt(X), C = £(F), K a field. A simple KX - module A such that Cx{A) = < 1 > there exists if and only if there exists a simple KC - module B such that Cc(B) = < 1 >. Proof We will suppose that X is non-abelian. We recall first that F is nilpotent and F = Cx([X,X]) [RD 1, Lemma 2.2.2]. Suppose that there exists a simple KXmodule A such that Cx(A) = < 1 >. Let 0 * a e A and D = O ( a ) . Since F is nilpotent, D is subnormal in X, and so D is normal in X. For every x e X,d e D we have xd = d\x for some element d\ e D; therefore (ax)d = a(xd) = a{d\x) = {ad\)x = ax. It follows that D < Cx(A) = < 1 >. In other words, CF(A) = < 1 >. Assume now that F is non-periodic; thus X is a soluble T - group of type I. From the description of these groups given at the beginning of this chapter, we obtain that F = C is abelian and X = < x, C > where cx = c _1 for all c e C, and x1 e C. In particular, the index \X : C\ is finite. By Proposition 3.6 A includes a simple KC - submodule B. We have already proved that CQ (B) = < 1 >. Now suppose that F is periodic. Letp e Tl(C),P be a Sylowp - subgroup of C, Pi = Qi(P). Clearly C < FC(G). By Lemma 3.8 p * charK and P\ includes a subgroup J such that \P\IJ\ = p and Cored-?) = < 1 >. Obviously, J is subnormal in G and therefore normal. It follows that J = CoredJ) = < 1 >. In other words, |Pi| = p so P is a Priifer p - subgroup. It follows that C is locally cyclic qlsubgroup where q = charK. By Corollary 2.4 and Theorem 2.6 there exists a
Groups whose Proper Factor-Groups Have a Transitive Normality Relation
201
simple KC - module B such that Cc{B) = < 1 >. Conversely, let there exists a simple KC - module B such that Cc(B) = < 1 >. Put U = B ®KC KX. Let A be a KX - composition factor of U. Let H be a non-identity normal subgroup of X and suppose that Hf) C = < 1 >. Since X is metabelian, HO [X,X] implies H< C. Thus Hp[ C * < 1 >. In particular, if CG(A) * < 1 >, then and CG{A) f l C = Cc(^4) * < 1 >. Using the arguments of Chapter 3, we can obtain a contradiction. Proposition 17.10 reduces the general situation of a simple module over soluble T - group Xto the case of a simple module over abelian group. This case has been in detail considered in Chapter 2. Consider also some partial cases. 17.11. Lemma [RD 11] A finitely generated soluble just non-T-group is finite. Proof Suppose the contrary, let G be a finitely generated just non-r-group which is infinite. First of all observe that G cannot be nilpotent. Indeed, a nilpotent just non-r-group is just non-abelian by Theorem 17.3, and Theorem 11.2 implies that G must be periodic and therefore finite. Let A be a non-identity normal abelian subgroup of G. Suppose that GIA is infinite. If B be a non-identity G - invariant subgroup of A, then GIB is abelian and [G,G] < B. Hence [G,G] is a minimal normal subgroup of G and [G,G] < A. Therefore GICG([G,G]) is a finitely generated abelian group. It follows that [G, G] is a finite elementary abelian p - subgroup for some prime p (see Corollary 1.17 and Corollary 2.2). Put C = CG([G,G]), then GIC is finite by Corollary 2.2. Also [C,C] < [G,G] < £(C); thus for x,y e C we have 1 = [x,y]p = [xp,y]. This means that C < f(C), in particular, C is abelian. Since GIC is periodic, it is finite. Thus Cp is a finitely generated (see, for example, [RD 9, Theorem 1.41]). There is a number n e N such that H = (Cp)n is torsion-free. Note that GIH is finite. This contradicts Lemma 17.2. Therefore GIA is finite. Then A is finitely generated and infinite. Therefore without loss of generality we can assume that A is a free abelian. Let L = [[G,G],G], observe that I * < 1 >. I f l n ^ * < l > , then G/(L (~)A) is finite, by the first part of the proof, and, replacing A by A f] L, we may assume that A < L. Let p e U(GIL); then GIAP is a finite soluble T group; however/? e H(GIL) C\ Tl(LIAp), which is impossible [GW]. Thus Af\L = < 1 > and L = LAI A, in particular, L is finite and abelian; and hence GIL is infinite. But this situation has been shown to be impossible. 17.12. Theorem [RD 11] A finitely generated hyperabelian group G which is not a T - group has a finite factor-group which is not a T - group. Proof. Suppose that
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Just non-X-Groups
=Ho
