Implications of the index of a fixed point subgroup

Abstract

Let $G$ be a finite group and $A\leq \operatorname{Aut}(G)$. The index $|G\colon C\_G(A)|$ is called the index of A in G and is denoted by Ind$\_G(A)$. In this paper, we study the influence of Ind$\_G(A)$ on the structure of $G$ and prove that $\[G,{} A]$ is solvable in case where $A$ is cyclic, Ind$\_G(A)$ is squarefree and the orders of $G$ and $A$ are coprime. Moreover, for arbitrary $A\leq \operatorname{Aut}(G)$ whose order is coprime to the order of $G$, we show that when $\[G,A]$ is solvable, the Fitting height of $\[G,A]$ is bounded above by the number of primes (counted with multiplicities) dividing Ind$\_G(A)$ and this bound is best possible.