On the conjecture of non-inner automorphisms of finite p-groups with a non-trivial abelian direct factor

2025/03/02 by Mandeep Singh, Singh, Mandeep, Mahak Sharma +1
Computer Science · Mathematics · #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Primary: 20D15 #Secondary: 20D45

paper · pdf · doi:10.48550/arxiv.2503.00954

openalex publication_date 2025/03/02 · openalex created_date 2025/10/12 · openalex updated_date 2026/07/28

Abstract

Let p be a prime number. A longstanding conjecture asserts that every finite non-abelian p-group has a non-inner automorphism of order p. In this paper, we prove that the conjecture is true when a finite non-abelian p-group G has a non-trivial abelian direct factor. Moreover, we prove that the non-inner automorphism is central and fixes Φ(G) elementwise. As a consequence, we prove that every group which is not purely non-abelian has a non-inner central automorphism of order p which fixes Φ(G) elementwise.

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