Varieties and simple groups

Abstract

In her book on varieties of groups, Hanna Neumann posed the following problem [13, p. 166]: “Can a variety other than D contain an infinite number of non-isomorphic non-abelian finite simple groups?”The answer to this question does not seem to be known at present. However, in [7], Heineken and Neumann described an algorithm for determining whether or not there are any non-abelian finite simple groups satisfying a given law. They also outlined a way in which their algorithm could be used to show that “only finitely many of the known non-abelian finite simple groups can satisfy a given non-trivial law”; in this paper, we shall follow their suggestions, and prove theTHEOREM. Let g be a set of mutually non-isomorphic non-abelian finite simple groups, each of which is either an alternating group or a group of Lie type, and let g generate a proper subvariety of D. Then y is finite.

Diameters of Finite Simple Groups: Sharp Bounds and Applications

“…By a result of Jones [16], every proper variety of groups contains only finitely many finite simple groups. This means that, if w = 1, then w(G) = {1} for all large enough finite simple groups G. Proof.…”

On the shortest identity in finite simple groups of Lie type

“…In the book [15], Hanna Neumann asked ( [15, p. 166]) whether there is a law which is satisfied in an infinite number of non-isomorphic non-abelian finite simple groups. Jones [8] gave a negative answer to this question, but his proof gives no explicit bound on the length of the shortest identity in each family of finite simple groups of Lie type (except for the case of Suzuki groups). Our work can be thought of as a quantitative version of Jones's results.…”

“…For every n A N, SL 2 ðKÞ has a subgroup isomorphic to SL 2 ðq n Þ, hence we obtain a law for the infinite family fSL 2 ðq n Þg y n¼1 . But the main theorem of Jones [8] states that there is no law which is satisfied by an infinite family of non-abelian finite simple groups, a contradiction. Now let us consider Case (ii).…”

Group Laws Implying Virtual Nilpotence

“…It is well known that A/B terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1446788700003335 [7] Group laws implying virtual nilpotence 301 decomposes as a direct product Si x S 2 x • • • S; of finite non-abelian simple groups all isomorphic to the same simple group 5, and that each element g of G acts by conjugation on A/B in such a way as to permute the S,; write n g for the permutation of {Si,... , Si) induced by conjugation by g. Write c = c(x, y) for the right-hand side of the law (4), a consequence of w = 1. We first show that for each g e G if any orbit under the above action of g on {Si,... , Si) has size exceeding 2R, then for every member S r of that orbit we have c(hB, gB) = 1 in A/B for all hB e S r .…”

“…By Jones [7] only the variety of all groups contains infinitely many pairwise nonisomorphic finite non-abelian simple groups. Hence in particular there are only a finite number of pairwise non-isomorphic groups that can occur as composition factors of groups satisfying any nontrivial law w = 1 of length < N having the same property as w = 1.…”