Bounding the order of complex linear groups and permutation groups with selected composition factors

TL;DR

This paper establishes bounds on the size of finite subgroups of complex linear groups with specific composition factors, motivated by tensor category growth questions, and explores broader applications.

Contribution

It provides new bounds on the order of finite subgroups of GL(n,C) with restricted composition factors, extending previous understanding.

Findings

Derived bounds for subgroup orders with certain composition factors

Connected bounds to growth rates in tensor categories

Applied results to broader group-theoretic contexts

Abstract

Originally motivated by questions of P. Etingof related to growth rates of tensor powers in symmetric tensor categories, we obtain general bounds on the order of finite subgroups of ${\rm GL}(n,\mathbb{C})$ with restricted composition factors (as usual, modulo Abelian normal subgroups). We place the question in a wider context, and find other applications.