2.
K. A. Zhevlakov, "Alternative rings," Algebra Logika, No. 3, 11-36 (1966); No. 4, 113117 (1967). 3. I. M. Mikheev, "...
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2.
K. A. Zhevlakov, "Alternative rings," Algebra Logika, No. 3, 11-36 (1966); No. 4, 113117 (1967). 3. I. M. Mikheev, "Locally right-nilpotent radical in the class of right-alternative rings," Algebra Logika, i_~i,No. 2, 174-185 (1972). 4. I. M. Mikheev, "Simple right-alternative rings," Algebra Logika, 16, No. 6, 682-711 (1977). 5. R. E. Roomel'di, "Nilpotency of ideals in (--I, 1)-rings with the minimality condition," Algebra Logika, 12, No. 3, 333-348 (1973). 6. E. I. Zel'manov and V. G. Skosyrskii, "Special Jordan algebras of nil-bounded index," Algebra Logika, 22, No. 6, 626-636 (1983). 7. E. I. Zel'manov, "A characterization of the McCrimmon radical," Sib. Mat. Zh., 25, No. 5, 190-192 (1984). 8. E. I. Zel'manov, "On prime Jordan algebras. II," Sib. Mat. Zh., 24, No. I, 89-105 (1983). 9. V. G. Skosyrskii, "On nilpotency in Jordan and right-alternative algebras," Algebra Logika, 18, No. i, 73-85 (1979). i0. V. G. Skosyrskii, " R ight-alternatlve algebras, " Algebra Logika, 23 No. 2, 185-192 (1984). II. A. Thedy, "Ri ght-alternatlve rings," J. Algebra, 37, 1-43 (1975). 12. A. Thedy, "Right-alternative rings with minimal condition," Math. Z., 155, No. 3, 277286 (1977). 13. A. Thedy, "Radicals of right-alternative and Jordan rings," Commun. Algebra, i_~2, 857887 (1984). •
•
,
2-GENERATOR GOLOD p-GROUPS A. V. Timofeenko
UDC 519.45
For every ~ ~ ~ and every field ~ , Golod [1] came up with a construction of a nonnilpotent (infinite-dimensional) ~-generator algebra V over the field ~ , such that every subalgebra with ~ - i generators is nilpotent. Golod's construction yields V in the form q / X , where Q is the free associative algebra of polynomials without constant term in the noncommuting indeterminates ~ , ' ' ' , ~ A over the field ~, and ~ is a homogeneous ideal (i.e., an ideal generated by homogeneous ~olynomials) in 0 • The indicated construction depends significantly on the following condition, to be called the Golod condition, that guarantees that the algebra V "- ~ I X will be infinitedimensional whenever all the coefficients of the inverse of the series
I- g¢ + are nonnegative, where $ ~ ~ generating set of the ideal~ If the algebra V----- Q / ~ algebra.
Let p----- ~ [ p ~
t
and ~ denotes the number of polynomials of degree ~ in a [2, Lemma 3 and Remark]; see also [3, Theorem 26.2.2]. satisfies the Golod condition, it is usually called a Golod
, ? a prime number and V a Golod nil-algebra.
of the multiplicative p-group It is not hard to verify that ~
4+V
The subgroup
is usually called a Golod group.
is an infinite
~-generator
p -group, and if all subalge-
brag with ( ~ - ~ )-generator subgroups in V are nilpotent, then all ( ~ subgroups of ~ are finite [3, Example 18.3.2].
~)-generator
Translated from Algebra i Logika, Vol. 24, No. 2, pp. 211-225, March-April, 1985. Original article submitted May 12, 1984.
0002-5232/85/2402-0129509.50
© 1986 Plenum Publishing Corporation
129
In the present paper (see the proof of Theorem i and Remark 3), we present a method for finding a finitely generated inflnite-dimensional subalgebra of a Golod algebra, which enables us to do the following. i) We can construct a finitely generated nonnilpotent nil-algebra the Golod condition (the example in Sec. 2).
that fails to satisfy
It is meaningful, in this sense, to talk of
a generalization of Golod's result. 2) In connection with V. P. Shunkov's question 6.58 in Kourovskaya Tetrad'
[4], we can
find (Theorem 4) infinite subgroups inside a 2-generator Golod p -group, each generated by a pair of conjugate elements of order an odd prime. the class of 2-generator Golod p -groups ( ~ > ~
Of course, this does not
that in
mean
there are no conjugate biprimitively finite
groups (for the definition, see Sec. 3). First of all, we will prove Theorem 1 for ~ = [ .
The construction of the algebra A
of Theorem 1 was carried out Jointly with V. P. Shunkov for all
i.
~)
~ •
Known Facts, Definitions, and Auxiliary Propositions Let ~
be the free associative algebra of polynomials without constant term in the in-
the subalgebra of F generated by the monomials % .... , ~ Remark I.
with ~ ) [.
The generators we chose for the algebra ~i
have the following properties
(*) and (**): 0
(*) if ~ = ~(~,~)is a monomlal, ~
~and~-~[~
...., ~
then every word in this alphabet that is distinct from $, i.e., V~
$0[~,...~,
.
is also distinct
is the free associative algebra of polynomials without constant term in the
free variables
~i,.,.,~i;
(**) if ~4' ~i
are monomials, one of which has degree distinct from 0, and ~ , ~ £ E
then for every monomial ~ ~ Vl Remark 2. ~"
word in the alphabet ICb 'f
the polynomial
%ii%E
~i '
Vt.
We will adopt the convention that if some polynomial ~ ~ V~ in the variables
is to be considered as a polynomial in the variables
we see, for example, that degree " ~ _---( ~ + ~
-degree
--
~4,...,~5, , then we mark $ with
~.o
Each paragraph that follows will be numbered, and we will refer to the information contanned in the paragraph as a proposition bearing the same number. I. number ~
The Golod condition will be fulfilled if the following restriction applies to the of monomlals of degree ~ in a generating set for the ideal
~ :
(i) where 6 denotes n o w - and for the rest of the article- some positive number ~ 2], see also [3, Example 26.2.3]. 130
J
[I, Lemma
2.
Let ~ be the homogeneous ideal of Golod in the free associative algebra Q
nomials without constant term, in ~
indetermlnates, over the field
resulting infinlte-dimenslonal algebra, e a c h ~ - 4 ~ potent.
p
of poly-
, and let Q/~ be the
-generator subalgebra of which is nil-
The homogeneous polynomials that generate the ideal X then possess the following
properties: a) their degrees run through all thenumbers
~i, N~, Nz+4~., .,~NE,
N~,...~N~, N~+4,
N~+Z,..,~N~, N~+~,... ; b) the number of polynomials of degrees N~,
N +4,...,~N