# Implications of the index of a fixed point subgroup

### Erkan Murat Türkan

Çankaya University, Ankara, Turkey and Middle East Technical University, Ankara, Turkey

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## 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.

## Cite this article

Erkan Murat Türkan, Implications of the index of a fixed point subgroup. Rend. Sem. Mat. Univ. Padova 142 (2019), pp. 1–7

DOI 10.4171/RSMUP/26