Implications of the index of a fixed point subgroup

Türkan, Erkan Murat

Rendiconti del Seminario Matematico della Università di Padova, Tome 142 (2019), pp. 1-7.

Let $G$ be a finite group and $A\le Aut\left(G\right)$. The index $|G:C_G\left(A\right)|$ is called the index of A in G and is denoted by Ind${}_G\left(A\right)$. In this paper, we study the influence of Ind${}_G\left(A\right)$ on the structure of $G$ and prove that $[G,A]$ is solvable in case where $A$ is cyclic, Ind${}_G\left(A\right)$ is squarefree and the orders of $G$ and $A$ are coprime. Moreover, for arbitrary $A\le Aut\left(G\right)$ 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${}_G\left(A\right)$ and this bound is best possible.

Classification : 20, 00
Mots clés : Index, fixed point subgroup, automorphism of a group, solvable group, fitting height