Characterizations of some groups in terms of centralizers

2021/09/22 by Sekhar Jyoti Baishya, Baishya, Sekhar Jyoti
Mathematics · Neuroscience · Social Sciences · #FOS: Mathematics #Finite Group Theory Research #Global Educational Reforms and Inequalities #Group Theory (math.GR) #Nuclear Receptors and Signaling

paper · pdf · doi:10.48550/arxiv.2109.10530

openalex publication_date 2021/09/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A group G is said to be n-centralizer if its number of element centralizers | \Cent(G)|=n, an F-group if every non-central element centralizer contains no other element centralizer and a CA-group if all non-central element centralizers are abelian. For any non-abelian n-centralizer group G, we prove that | (G)/(Z(G))| ≤ (n-2)², if n ≤ 12 and | (G)/(Z(G))| ≤ 2(n-4)^log₂^(n-4) otherwise, which improves an earlier result. We prove that if G is an arbitrary non-abelian n-centralizer F-group, then gcd(n-2, | (G)/(Z(G))|) ≠ 1. For a finite F-group G, we show that | \Cent(G)| ≥ (| G |)/(2) iff G ≅ A₄ , an extraspecial 2-group or a Frobenius group with abelian kernel and complement of order 2. Among other results, for a finite group G with non-trivial center, it is proved that | \Cent(G)| = (| G | )/(2) iff G is an extraspecial 2-group. We give a family of F-groups which are not CA-groups and extend an earlier result.

Related

• On n-centralizer CA-groups
• Counting the Number of Centralizers of 2-Element Subsets in a Finite Group
• Counting Centralizers of a Finite Group with an Application in Constructing the Commuting Conjugacy Class Graph
• On finite groups with nine centralizers
• Counting centralizers and z-classes of some F-groups