Finite groups with some particular maximal invariant subgroups being nilpotent or all non-nilpotent maximal invariant subgroups being normal

TL;DR

This paper classifies finite groups with specific maximal invariant subgroups under automorphisms, showing equivalences between certain nilpotency and normality conditions, advancing understanding of group structure under automorphism actions.

Contribution

It provides a complete classification of finite groups where maximal invariant subgroups containing normalizers of Sylow subgroups are nilpotent, and proves the equivalence of key nilpotency and normality hypotheses.

Findings

Complete classification of such finite groups.

Equivalence of nilpotent and normal maximal invariant subgroup conditions.

Insight into the structure of groups with automorphism actions.

Abstract

Let $A$ and $G$ be finite groups such that $A$ acts coprimely on $G$ by automorphisms. We provide a complete classification of a finite group $G$ in which every maximal $A$-invariant subgroup containing the normalizer of some $A$-invariant Sylow subgroup is nilpotent. Moreover, we show that both the hypothesis that every maximal $A$-invariant subgroup of $G$ containing the normalizer of some $A$-invariant Sylow subgroup is nilpotent and the hypothesis that every non-nilpotent maximal $A$-invariant subgroup of $G$ is normal are equivalent.