# On the length of finite groups and of fixed points

Research paper by **E. I. Khukhro, P. Shumyatsky**

Indexed on: **29 Jan '15**Published on: **29 Jan '15**Published in: **Mathematics - Group Theory**

Join Sparrho today to stay on top of science

Discover, organise and share research that matters to you

Join Sparrho today to stay on top of science

Discover, organise and share research that matters to you

Join for free

#### 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.