Lower bounds on the maximum dimension of a simple module in characteristic p

TL;DR

This paper establishes lower bounds on the maximum dimension of simple modules over finite groups in characteristic p, with special cases for p=2 or Mersenne primes, based on subgroup properties.

Contribution

It provides new lower bounds for simple module dimensions in characteristic p, accounting for special primes like 2 and Mersenne primes, and relates bounds to subgroup properties.

Findings

Lower bounds depend on p-subgroup properties.

Special bounds are derived for p=2 and Mersenne primes.

Results improve understanding of module dimensions in modular representation theory.

Abstract

We obtain lower bounds for the maximum dimension of a simple FG-module, where G is a finite group and F is an algebraically closed field of characteristic p. The bounds are described in terms of properties of p-subgroups of G. When p is 2 or p is a Mersenne prime, the bounds take a different form, due to exceptions which arise for such primes.