2-Generator conditions in linear groups

Vol. XXII, 1971

237

2-Generator Conditions in Linear Groups By B. A. F. WEm~FRITZ

In [4] V. P. PLATONOV states "a linear group over a field of characteristic zero in which every 2-generator subgroup is soluble-by-finite is itself soluble-by-finite". His proof contains a flaw since the 2-variable law he produces contains more than two variables. The proof can be repaired since a law of the required type must exist; the class of all groups each of whose 2-generator subgroups lies in the product of the variety of soluble groups of derived length at most r and the variety generated by the symmetric group on s symbols is a variety ([2] 16.21) which trivially is not the variety of all groups. PLATOI~OV'S proof relies on a relatively deep result of CHEVALLEY ([1] Prop. 23.2). We give another proof (in fact of a little more) but use even deeper results of J. G. THOMPSON, namely the classification of the (finite) minimal simple groups. I t follows from this classification that every such group is 2-generator. We then give two proofs of a nilpotent analogue of PLATONOV'S result which turns out to be a little more subtle. Lemma 1. Let R be a finitely generated integral domain and G a subgroup o / G L (n, R) such that every 2-generator subgroup o/G is soluble-by-/inite. Then G is soluble-by-/inite.

P r o o f * ) . G contains a normal subgroup H of finite index such that H is residually nilpotent ([6] 4.7). I t follows from [5] 6.25 that a residually nilpotent group is soluble ff it is soluble-by-finite. Hence every 2-generator subgroup of H is soluble and consequently every finite image of H is soluble by T ~ o ~ r s o ~ ' s results. B y MXL'CEV'S Theorem, [6] 4.2, H is residually linear of degree n and thus is residually soluble of bounded degree, [6] 3.7. Therefore H is soluble. Throughout F denotes a (commutative) field. Lemma 2. Let G be a subgroup o/GL (n, F) and ~ a variety o/groups such that every /initely generated subgroup o/G is a/inite extension o] a ~-grou p. Then G is an extension o / a ~-group by a periodic linear group over t". P r o of. Let V be a ~-subgroup ofGL (n, iv) then ~r (V), the closure of V in GL (n, F), is also a ~-group. Hence if H is any finitely generated subgroup of G then Z~CF(H 0) = = d F (H) ~ the connected component of ~r (H) containing 1, is a !3-group. Pick a *) It is also possible to adapt PLATO~OV'Smethod to prove this lemma.

238

B . A . F . WEHRFRITZ

ARCH. MATH.

finitely generated subgroup H of G such that d F (H) ~ has maximal dimension. For each g in G, d R ( H ) ~ ~ d F ( ( g , H ) ) ~ and thus S ----G t~ ~r is a closed normal g-subgroup of G. Also since d F ( H ) ~ h a s finite index in d F ( ( g , H}), some positive power of g lies in S and G/S is periodic. Finally G/S is isomorphic to a linear group over • ([6] 6.4). Theorem 1. I] G is a subgroup o] G L(n, F) such that every 2-generator subgroup o] G is soluble-by-]inite, then G is soluble-by-periodic and i/ char F - ~ 0 then G is even soluble.by-/inite. P r o o f . B y L e m m a 1 every finitely generated subgroup of G is soluble-by-finite and hence b y L e m m a 2, G is soluble-by-periodic. I f char F = 0 then every periodic linear group over /~ is abelian-by-finite (Schur's Theorem [6] 9.4) and the theorem follows. 9 We clearly need to restrict the characteristic for the last part of the theorem since there exist infinite simple periodic linear groups. The following corollary is an immediate consequence of the theorem and [6] 10.3. Corollary. Let R be a Noetherian (commutative) ring, M a ]initely generated R-module and G a group o/R-automorphisms o] M. I] every 2.generator subgroup o] G is solubleby-/inite then G is soluble-by-periodic. Theorem 2. Let G be a subgroup o] GL (n, P) and suppose that every 2-generator subgroup o/G is nilpotent-by-/inite. Then G is nilpotent-by.periodic. l~irst p r o o f . We m a y clearly assume t h a t F is algebraically closed. By Theorem 1, G is soluble-by-periodic, so we m a y suppose that G is triangularizable ([6] 3.6). Then G-----dF (G) is also triangularizable. Denote by V the maximal unipotent subgroup of G and put U ----G c~ V. Let H = (gl . . . . , gr) be a finitely generated subgroup of G. I f u e U then (u, g,) is nilpotent-by-finite and so ,~/F((U, g~})0 is nflpotent. Therefore (Vc~ d r ((u, g~))) ~ F ((u, g~))0 is also nitpotent. By [6] 7.3, (gi)a e ~r g~)) and in a nflpotent linear ~ o u p unipotent elements commute with d-elements ([6] 7.11). Hence for each u in U there exists an integer k (u) such t h a t for i : 1, 2, .. ., r we have [u, (~tild'-~k(,~)l,---- 1. Let K = ( V , (gi)a : i = 1, 2, ..., r) and for any integer k let K~ ----(V, ((gi)a) ~ : i = 1, 2 . . . . . r ) . Since G/V is abelian (K : K~) is finite for every/c. I f u e U then VC~(u) is 'closed in G; and clearly K~(u) C VCs(u). Therefore K ~ C VC~(u) for each u in U. Let I : (K : K0). Since (gi)u e V for each i we have H C K and HI C K 0. Hence [U, H~] C [U, V](~ G C U. Now V is unipotent and so is nilpotent of class at most m = (n - - 1 ) . Therefore [U,mH Z] C [U,mV] = {1} and H l is nilpotent of class at most m ~ 1 (in fact at most m by [6] 7.11, unless n ~- 1). But H is soluble and finitely generated, so H / H 1 is finite. Thus every finitely generated subgroup of G is a finite extension of a nilpotent group of class at most m ~- 1, and the conclusion of the theorem follows from L e m m a 2.

Vol. XXII, 1971

Linear Groups

239

S e c o n d p r o o f . Again by Theorem 1 we may assume that G is triangularizable, so in particular G is nilpotent-by-abelian. Let H = (gl . . . . , gr> be a finitely generated subgroup of G and denote b y l the least common multiple of the r integers ((g~> : (gl>~ I f x e H then (gi, ~> is nilpotent-by-finite, so 0 is nilpotent. This latter group contains both is nilpotent-by-abelian, and thus the Engelizer in ( S ) of every element of (S> is a subgroup ([3] Corollary 1). But then the Engelizer in (S> of every element of S is <S> itself and consequently (S> is generated by left Engel elements. By a theorem o f K . W. GRVE~B~G ([6] 8.15) <S> is locally nflpotent and so nilpotent ([6] 4.17). I f U denotes the maximal unipotent subgroup of H then ( S ) U is a nilpotent normal subgroup of H and H / ( S > U is an abelian group that is generated by a finite number of elements of finite order. Therefore H is nilpotent-by-finite and the theorem follows from the triangularizabflity of H , [6] 4.13 and Lemma 2. Although both of these proofs use Lemma 1, Theorem 2 does not really depend upon the deep results of THO~rSON or CH~VALL~Y.That is, in the situation of Lemma 1, ff we assume that every 2-generator subgroup of G is nilpotent-by-t~nite, we may deduce that G is soluble-by-finite in an elementary way. For by [6] 4.7 we can choose H to be residually a finite p-group for some prime p and thus by [5] 6.25 every 2-generator subgroup of H is an extension of a nilpotent group by a finite p-group. Consider a finite homomorphie image /~ of H. The' subgroup generated by any two p'-elements o f / ~ is actually nilpotent and so is a p'-group. I t follows easily t h a t / / has a nilpotent normal p-complement and that /it is soluble. The proof is then completed as before. I f char 2' ~-- 0 it is not possible to conclude in Theorem 2 that G is nilpotent-byfinite. I t is easy to see that

/1 provides an appropriate counterexample. I do not know whether in Theorem 2, 2" can be replaced by a Noetherian ring as in the corollary to Theorem 1. In conclusion perhaps one should remark that it follows easily from [6] 10.13, 10.14 and 10.20 that a group of automorphisms of a finitely generated module over a commutative ring is hyperabelian (resp. locally nilpotent, in fact even a Gruenberg group, see [5] p. 99 for the definition) ff each of its 2-generator subgroups is soluble (resT. nilpotent). Further a linear group is hypercyelic if each of its 2-generator subgroups is supersoluble ([7] Corollary B2). References [1] [2] [3] [4]

C. C~EV~LL~Y,Classification des groupes de Lie alg6briques. S~m. ~cole Norm. Sup. 1956--58 H. NEUral-N, Varieties of Groups. Berlin 1967. T. A. P~No. On groups with nilpotent derived groups. Arch. Math. 20, 251--253 (1969). V. P. PLATO~OV,Several problems on linear groups. Mat. Zametki 4, 635--638 (1968) (Russian).

240

B . A . F . WEHP,FI~ITZ

A!ZCH.MATH.

[5] D. J. S. RoBI~SOI% Infinite Soluble and Nilpotent Groups. Queen Mary College Mathematics Notes (1968), [6] B. A. F. Wv,nmFl~i~rz, Infinite Linear Groups. Queen Mary College Mathematics Notes (1969). [7] B. A. F. W]~XRImlTZ, Supersoluble and locally supersoluble linear groups. J. Algebra 17, 41--58 (1971). Eingegangen am 29. 7. 1970 Ansehrift des Autors: B. A. F. Wehrfritz Mathematics Department Queen Mary College London E. 1, England