Groups of automorphisms with TNI-centralizers (Q683499)

A number of results give information about a finite group $G$, when we are given a finite group $A$ of automorphisms of $G$, and some further information. This information on $G$ is particularly strong when $C_G(A) = 1$. The authors obtain similar results when the condition that $C_G(A) = 1$ is weakened to the condition that $C_G(A)$ is a TNI-subgroup of $G$. A subgroup $H$ of a finite group $G$ is a TNI-subgroup if, for all $g \in G$ if $g \notin N_G(H)$ then $N_G(H) \cap H^g = 1$. For example, the authors show that, if $C_G(A)$ is a TNI-subgroup of $G$, then $G$ is solvable if and only if $C_G(A)$ is solvable. Furthermore they find some bounds on the nilpotent length of $G$ in terms of the nilpotent length of $C_G(A)$ under some further conditions. They also consider in more detail the case when $A$ is a Frobenius group.