On the influence of the fixed points of an automorphism to the structure\n of a group

2020/09/06 by M. Yasi̇r Kızmaz, Kızmaz, M. Yasir
Mathematics · Social Sciences · #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Migration, Ethnicity, and Economy

paper · pdf · doi:10.48550/arxiv.2009.02677

openalex publication_date 2020/09/06 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

Let \α be a coprime automorphism of a group G of prime order and let\nP be an \α-invariant Sylow p-subgroup of G. Assume that p\∉\n\π(C_G(\α)). Firstly, we prove that G is p-nilpotent if and only if\nC_NG(P)(\α) centralizes P. In the case that G is Sz(2^r) and\nPSL(2,2^r)-free where r=|\α|, we show that G is p-closed if and\nonly if C_G(\α) normalizes P. As a consequences of these two results,\nwe obtain that G\≅ P\× H for a group H if and only if C_G(\α)\ncentralizes P. We also prove a generalization of the Frobenius p-nilpotency\ntheorem for groups admitting a group of automorphisms of coprime order.\n

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