On Autonilpotent Finite Groups

TL;DR

This paper characterizes autonilpotent finite groups through their automorphism groups, establishing conditions for p-groups and extending classical theorems like Baer's theorem to this class.

Contribution

It introduces a new characterization of autonilpotent groups based on automorphism stabilizations and extends key theorems to this class.

Findings

A p-group is autonilpotent iff its automorphism group is also a p-group.

Established analogues of Baer's theorem for autonilpotent groups.

Derived criteria for hypercenter and Frobenius p-nilpotency in autonilpotent groups.

Abstract

In the paper autonilpotent groups were characterized as groups $G$ such that $\mathrm{Aut}G$ stabilizes some chain of subgroups of $G$. It was shown that a $p$-group is autonilpotent if and only if its group of automorphisms is also a $p$-group. Analogues of Baer's theorem about the hypercenter and Frobenius $p$-nilpotency criterion were obtained for autonilpotent groups.