Implications of the index of a fixed point subgroup

Abstract

Let $G$ be a finite group and $A\le \mathrm{Aut}(G)$. The index $|G:C_G(A)|$ 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 $A\le \mathrm{Aut}(G)$ 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.