Class-preserving automorphisms of some finite $p$-groups

TL;DR

This paper characterizes finite p-groups of order p^5 that admit non-inner class-preserving automorphisms, providing necessary and sufficient conditions and offering simplified proofs of existing results.

Contribution

It establishes precise conditions for the existence of non-inner class-preserving automorphisms in certain finite p-groups, advancing understanding of their automorphism structure.

Findings

Identifies conditions for non-inner class-preserving automorphisms in p^5 groups

Provides simplified proofs of previous theorems in the literature

Enhances understanding of automorphism groups of finite p-groups

Abstract

Let $G$ be a finite $p$-group of order $p^5$, where $p$ is a prime. We give necessary and sufficient conditions on $G$ such that $G$ has a non-inner class-preserving automorphism. As a consequence, we give short and alternate proofs of results of section 5 of Yadav [Proc. Indian Acad. Sci. (Math. Sci.) 118 (2008), 1-11] and Theorem 4.2 of Kalra and Gumber [Indian J. Pure Appl. Math. 44 (2013), 711-725].