Bounding the order of finite $p^{\prime}$- subgroups of ${\rm GL}(n,\mathbb{C})$

TL;DR

This paper establishes a uniform bound on the index of an Abelian normal subgroup within finite subgroups of GL(n,C) whose order is coprime to a prime p, with the bound depending on p and n.

Contribution

It proves a universal bound on the structure of p'-subgroups of GL(n,C), extending understanding of their composition and normal Abelian subgroups.

Findings

Existence of a fixed constant C for all n and p

Bound on index of Abelian normal subgroup as (Cp)^{n-1}

Applicable to all finite p'-subgroups of GL(n,C)

Abstract

In this note, we prove: \medskip \noindent {\bf Theorem A:} \emph{ There is a fixed constant $C$ such that for any positive integer $n$ and prime $p$, every finite subgroup $G$ of order coprime to $p$ of ${\rm GL}(n,\mathbb{C})$ has an Abelian normal subgroup $A$ with $$[G:A] \leq (Cp)^{n-1}.$$}