Comparing the order and the minimal number of generators of a finite irreducible linear group

TL;DR

This paper establishes an upper bound relating the order and minimal number of generators of finite irreducible linear groups, providing insights into their structural complexity and aiding automorphism group computations.

Contribution

It proves a bound on the product of the minimal number of generators and the logarithm of the group order for irreducible subgroups of GL(n,q), with estimated constants.

Findings

d(G) log |G| = O(n^2 log q) for irreducible subgroups G of GL(n,q)

Provides bounds useful for automorphism group computations

Estimates constants involved in the bounds

Abstract

We prove that $d(G) \log |G| = O(n^2 \log q)$ for irreducible subgroups $G$ of GL$(n,q)$, and estimate the associated constants. The result is motivated by attempts to bound the complexity of computing the automorphism groups of various classes of finite groups.