Finite groups determined by the number of element centralizers. Issue 9 (2nd September 2017) - Jafarian Amiri, Seyyed Majid; Amiri, Mohsen; Rostami, Hojjat
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].