Products of conjugacy classes and fixed point spaces

Abstract

We prove several results on products of conjugacy classes in finite simple groups. The first result is that for any finite non-abelian simple groups, there exists a triple of conjugate elements with product 1 which generate the group. This result and other ideas are used to solve a 1966 conjecture of Peter Neumann about the existence of elements in an irreducible linear group with small fixed space. We also show that there always exist two conjugacy classes in a finite non-abelian simple group whose product contains every nontrivial element of the group. We use this to show that every element in a non-abelian finite simple group can be written as a product of two rth powers for any prime power r (in particular, a product of two squares answering a conjecture of Larsen, Shalev and Tiep).

On the uniform spread of almost simple linear groups

“…As described in [24, Examples 2.1-2.9], it turns out that such a subgroup belongs to one of nine specific subgroup collections. In [23], Guralnick and Malle prove the following useful corollary. Theorem 2.12.…”

“…If we can choose s so that some power of gs has order r, where r is a primitive prime divisor of q e − 1 with e > n/2, then we can use the aforementioned results in [23,24] to restrict significantly the possible subgroups in M(gs).…”

“…Since (n, q 0 ) ∈ B we may assume q 0 ≥ 4. Here |C PGL 4 (q 0 ) (y)| = (q 4 0 − 1)/(q 0 − 1) and y 2 is irreducible, so each H ∈ M is irreducible and the possible types are given in (23). In addition, we note that if H ∈ M is a C 3 -subgroup of type GL 2 (q 2 ) then C G (y 2 ) = C H (y 2 ) and (y 2 ) G ∩ H = (y 2 ) H , so there is a unique such subgroup in M. By applying the relevant fixed point ratio estimates in Lemma 2.11 we deduce that…”

Equations in simple matrix groups: algebra, geometry, arithmetic, dynamics

“…In addition, Guralnick and Malle [53] have extended the aforementioned result (17) from [73] and proved the following variant of Thompson's Conjecture.…”

“…Larsen, Shalev and Tiep [73] proved that any such word is surjective on sufficiently large finite simple groups. By further recent results of Guralnick and Malle [53] and of Liebeck, O'Brien, Shalev and Tiep [81], some words of the form are known to be surjective on all finite simple groups. The particular case of surjectivity of these words on SL(2 ) was studied in [8] (see subsection 6.1).…”