On the length of finite groups and of fixed points

TL;DR

This paper establishes bounds on the generalized Fitting height and nonsoluble length of finite groups with automorphisms of coprime order, relating these bounds to fixed-point subgroups and prime factor counts.

Contribution

It proves that the generalized Fitting height and nonsoluble length of a finite group are bounded by those of fixed-point subgroups under automorphisms of coprime order, extending previous results.

Findings

Bound on generalized Fitting height in terms of fixed points

Bound on nonsoluble length in terms of fixed points

Special case result for automorphisms of prime order

Abstract

The generalized Fitting height of a finite group $G$ is the least number $h=h^*(G)$ such that $F^*_h(G)=G$, where the $F^*_i(G)$ is the generalized Fitting series: $F^*_1(G)=F^*(G)$ and $F^*_{i+1}(G)$ is the inverse image of $F^*(G/F^*_{i}(G))$. It is proved that if $G$ admits a soluble group of automorphisms $A$ of coprime order, then $h^*(G)$ is bounded in terms of $h^* (C_G(A))$, where $C_G(A)$ is the fixed-point subgroup, and the number of prime factors of $|A|$ counting multiplicities. The result follows from the special case when $A=\langle\varphi\rangle$ is of prime order, where it is proved that $F^*(C_G(\varphi ))\leqslant F^*_{9}(G)$. The nonsoluble length $\lambda (G)$ of a finite group $G$ is defined as the minimum number of nonsoluble factors in a normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. It is proved that if $A$ is a group of automorphisms of $G$ of coprime order, then $\lambda (G)$ is bounded in terms of $\lambda (C_G(A))$ and the number of prime factors of $|A|$ counting multiplicities.