On the product decomposition conjecture for finite simple groups

Abstract

Abstract. We prove that if G is a finite simple group of Lie type and S a subset of G of size at least two then G is a product of at most c log |G|/ log |S| conjugates of S, where c depends only on the Lie rank of G. This confirms a conjecture of Liebeck, Nikolov and Shalev in the case of families of simple groups of bounded rank. We also obtain various related results about products of conjugates of a set within a group.

Quasirandom Group Actions

“…Both of these conjectures are proved for groups of Lie type of bounded rank [7,12,24]. We are able to give partial results for groups of Lie type of unbounded rank that complement those already in the literature due to the original Gowers trick.…”

A generalization of a theorem of Rodgers and Saxl for simple groups of bounded rank

“…The other is a result of Gill, Pyber, Short and Szabó [2]: Theorem 3 differs to that of Rodgers and Saxl in three important respects, two good, one not so good: First, our result pertains to all finite simple groups G of Lie type. Second, our result does not just pertain to conjugacy classes, but to subsets of the group, provided we are free to take conjugates.…”

“…The next result was obtained independently in [4, 14]. The subsequent corollary is an easy consequence, and can be found in [2]. Note that, by the translate of a set in a group , we mean a set of form where is some element of .…”

“…Note that, if we write G r (q) for a simple group of Lie type of rank r over F q , then there are multiple conventions for the definition of r and the definition of q. We have stated the following results very conservatively -they are valid for whichever standard definition of these two parameters one cares to take (and this also explains the difference in the statement of the first, from that which appears in [2]).…”

“…In this section, we use a variant of the 'greedy lemma' argument of [2]. First we need an easy little lemma.…”