On the length of finite groups and of fixed points

Abstract

The generalized Fitting height of a finite group $G$ is the least number $h=h^{\ast }(G)$ such that $F_h^{\ast }(G)=G$, where the $F_i^{\ast }(G)$ is the generalized Fitting series: $F_1^{\ast }(G)=F^{\ast }(G)$ and $F_{i+1}^{\ast }(G)$ is the inverse image of $F^{\ast }(G/F_i^{\ast }(G))$. It is proved that if $G$ admits a soluble group of automorphisms $A$ of coprime order, then $h^{\ast }(G)$ is bounded in terms of $h^{\ast }(C_G(A))$, 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 $A=⟨\phi ⟩$ is of prime order, where it is proved that $F^{\ast }(C_G(\phi ))⩽F_9^{\ast }(G)$. 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.

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