Effective Results on the Waring Problem for Finite Simple Groups

TL;DR

This paper establishes new results on the Waring problem for finite simple groups, showing that elements can be expressed as products of powers and conjugacy class products, with effective bounds for large groups.

Contribution

It provides an effective version of a key result on expressing elements as products of powers in finite simple groups, with explicit bounds and conditions.

Findings

Every non-central element is a product of two mth powers in large enough groups.

The verbal width of the mth power word is bounded by a function g(m).

Products of conjugacy classes cover all non-central elements in large groups.

Abstract

Let G be a finite quasisimple group of Lie type. We show that there are regular semisimple elements x,y in G, x of prime order, and |y| is divisible by at most two primes, such that the product of the conjugacy classes of x and y contain all non-central elements of G. In fact in all but four cases, y can be chosen to be of square-free order. Using this result, we prove an effective version of one of the main results of Larsen, Shalev and Tiep by showing that, given any positive integer m, if the order of a finite simple group S is at least f(m) for a specified function f, then every element in S is a product of two mth powers. Furthermore, the verbal width of the mth power word on any finite simple group S is at most g(m) for a specified function g. We also show that, given any two non-trivial words v, w, if G is a finite quasisimple group of large enough order, then v(G)w(G) contains all non-central elements of G.