On abelian group actions with TNI-centralizers

TL;DR

This paper investigates the structure of finite groups with abelian automorphism groups having TNI-centralizers, establishing bounds on the group's Fitting length based on properties of the centralizer and automorphism group.

Contribution

It introduces new bounds on the Fitting length of finite groups with abelian automorphism groups possessing TNI-centralizers, linking group structure to automorphism properties.

Findings

G is solvable with Fitting length at most h(C_G(A)) + ℓ(A)

If C_G(A) is nonnormal, then h(G) ≤ ℓ(A) + 3

Provides structural insights into groups with TNI-centralizers and abelian automorphisms

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.