Growth in Linear Algebraic Groups and Permutation Groups: Towards a Unified Perspective

Abstract

By now, we have a product theorem in every finite simple group G of Lie type, with the strength of the bound depending only in the rank of G. Such theorems have numerous consequences: bounds on the diameters of Cayley graphs, spectral gaps, and so forth. For the alternating group Altn, we have a quasipolylogarithmic diameter bound (Helfgott-Seress 2014), but it does not rest on a product theorem.We shall revisit the proof of the bound for Altn, bringing it closer to the proof for linear algebraic groups, and making some common themes clearer. As a result, we will show how to prove a product theorem for Altn -not of full strength, as that would be impossible, but strong enough to imply the diameter bound.

An improved diameter bound for finite simple groups of Lie type

“…The inductive proof of Helfgott and Seress relies heavily on the fact that their result extends to transitive groups. For a greatly simplified argument, see .…”

The diameter of products of finite simple groups

“…A c c e p t e d m a n u s c r i p t D. Dona: The diameter of products of finite simple groups Indeed, the general approach that we follow in our proof owes its validity to [8,Thm. 1.6], although we do not explicitly use the statement of that theorem: rather, we closely follow the proof of [5,Lemma 4.13] and show that the same reasoning applies to groups of Lie type as well. The way that the lemma is related to Liebeck-Shalev is through the use of the fact that every element in Alt(m) is a commutator ([5, Lemma 4.12], first proved in [9, Thm.…”

“…Dependence on the maximum of the diameter of the components, instead of dependence on their product as Schreier's lemma (see Lemma 2.1) would naturally give us, was already established in [2,Lemma 5.4]: in that case, the diameter was bounded as O(d 2 ), where the dependence of the constant on n was polynomial as in our statement. This result was improved in [5,Lemma 4.13] to O(d), but only in the case of alternating groups: this was done in part to fix a mistake in the use of the previously available result in Babai-Seress, which is why only alternating groups were considered, as permutation subgroups were the sole concern in both papers; a suggestion by Pyber, reported in Helfgott's paper, points at the results by Liebeck and Shalev [8] as a way to prove a bound of O(d) for a product of arbitrary non-abelian finite simple groups.…”

“…A more modest question is that of producing bounds for the diameter of direct products of finite simple groups, depending on the diameter of their factors. This is not an idle question, for bounds of this sort have been used more than once as intermediate steps towards the proof of bounds for simple groups themselves: Babai and Seress have done so in [2, Lemma 5.4], as well as Helfgott more than two decades later in [5,Lemma 4.13]. We improve on both results in the following theorem, which also features explicit constants.…”

“…The result of part (a) is known and elementary: see [2, Lemma 5.2], where the constant is marginally worse only due to the fact that sets of generators are not required to be symmetric (cfr. also [5,Lemma 4.14], which treats the case of G = (Z/pZ) e under this assumption). Part (c) is quite natural, given the different (in some sense, opposite) behaviour of abelian and non-abelian factors, as it can be readily observed in its short proof.…”