Cosets of Sylow p-subgroups and a Question of Richard Taylor

TL;DR

This paper demonstrates the existence of infinitely many finite simple groups with specific coset properties related to Sylow p-subgroups, addressing a question in representation theory and extending previous results for p=2.

Contribution

It generalizes Thompson's 1967 result for p=2 to all primes p, and applies this to answer Richard Taylor's question on adequate representations.

Findings

Existence of infinitely many finite simple groups with cosets where all elements have order divisible by p

Extension of Thompson's result from p=2 to all primes p

Application to questions in representation theory

Abstract

We prove that for any prime p there exist infinitely many finite simple groups G with a coset xP of a Sylow p-subgroup P of G such that every element of xP has order divisible by p. John Thompson proved this for p=2 in 1967 answering a question of Lowell Paige. This result is used to answer a question of Richard Taylor on adequate representations.