Finite groups determined by the number of element centralizers. Issue 9 (2nd September 2017)

• Record Type: Journal Article
• Authors: Jafarian Amiri, Seyyed Majid; Amiri, Mohsen; Rostami, Hojjat
• Is Part Of: Communications in algebra. Volume 45:Issue 9(2017)
• Page Start: 3792
• Page End: 3797
• Publication Date: 2017-09-02
• Subjects: Finite group -- minimal nonabelian group -- primitive n-centraliser group; 20D60; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2016.1246664
• Languages: English
• ISSNs: 0092-7872

Abstract

ABSTRACT: For a finite group G, let | Cent ( G )| and ω ( G ) denote the number of centralizers of its elements and the maximum size of a set of pairwise noncommuting elements of it, respectively. A group G is called n -centralizer if | Cent ( G )| = n and primitive n -centralizer if| C e n t ( G ) | = | C e n t ( G Z ( G ) ) | = n . In this paper, among other results, we find | Cent ( G )| and ω ( G ) whenG Z ( G ) is minimal nonabelian and this generalizes some previous results. We give a necessary and sufficient condition for a primitive n -centralizer group G with the minimal nonabelian central factor. Also we show that ifG Z ( G ) ≅ S 4, then G is a primitive 14-centralizer group and ω ( G ) = 10 or 13. Finally we confirm Conjecture 2.4 in [A. R. Ashrafi, On finite groups with a given number of centralizers, Algebra Colloq. 7(2) (2000), 139-146].