Automorphisms of semidirect products fixing the non-normal subgroup

TL;DR

This paper characterizes the automorphism groups of certain semidirect products, focusing on those fixing a non-normal subgroup, with explicit computations for specific non-abelian p-groups.

Contribution

It provides a detailed description of automorphisms fixing a non-normal subgroup in semidirect products, including explicit results for particular non-abelian p-groups.

Findings

Automorphism groups are explicitly computed for specific non-abelian p-groups.

Automorphisms fixing the non-normal subgroup are characterized.

Results extend understanding of automorphisms in semidirect product structures.

Abstract

In this paper, we describe the automorphism group of semidirect product of two groups that fixes the non-normal subgroup of it. We have computed these automorphisms for the non-abelian metacyclic $p$-group and non-abelian $p$-groups $(p\ge 5)$ of order $p^{4}$, where $p$ is a prime.