Products of normal subsets and derangements

TL;DR

This paper investigates the conditions under which products of normal subsets cover all non-trivial elements in finite simple groups, and shows that large simple groups of Lie type have this property, while others do not.

Contribution

It establishes criteria for product coverage in finite simple groups and proves that large simple groups have elements as products of two derangements in permutation representations.

Findings

Finite simple groups of Lie type of bounded rank satisfy the product property.

Alternating groups and groups with fixed q do not satisfy the property as n grows.

Every element in a large simple group's permutation representation is a product of two derangements.

Abstract

In recent years there has been significant progress in the study of products of subsets of finite groups and of finite simple groups in particular. In this paper we consider which families of finite simple groups $G$ have the property that for each $\epsilon > 0$ there exists $N > 0$ such that, if $|G| \ge N$ and $S, T$ are normal subsets of $G$ with at least $\epsilon |G|$ elements each, then every non-trivial element of $G$ is the product of an element of $S$ and an element of $T$. We show that this holds in a strong sense for finite simple groups of Lie type of bounded rank, while it does not hold for alternating groups or groups of the form ${\mathrm{PSL}}_n(q)$ where $q$ is fixed and $n$ tends to infinity. Our second main result is that any element in a transitive permutation representation of a sufficiently large finite simple group is a product of two derangements.