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

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 $$[G:A] \leq (Cp)^{n-1}.$$}