On abelian group actions with TNI-centralizers

Abstract

A subgroup $H$ of a group $G$ is said to be a TNI-subgroup if $N_{G}(H)\cap H^g=1$ for any $g\in G\,\backslash \,N_{G}(H).$ Let $A$ be an abelian group acting coprimely on the finite group $G$ by automorphisms in such a way that $C_G(A)=\{g\in G : g^a=g $\, for all $a\in A\}$ is a solvable TNI-subgroup of $G$. We prove that $G$ is a solvable group with Fitting length $h(G)$ is at most $h(C_G(A))+\ell(A)$. In particular $h(G)\leq \ell(A)+3$ whenever $C_G(A)$ is nonnormal. Here, $h(G)$ is the Fitting length of $G$ and $\ell(A)$ is the number of primes dividing $A$ counted with multiplicities.