Finite groups determined by the number of element centralizers

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 $\left|Cent\right(G\left)\right|=\left|Cent\right(\frac{G}{Z\left(G\right)}\left)\right|=n$. In this paper, among other results, we find |Cent(G)| and ω(G) when $\frac{G}{Z\left(G\right)}$ 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 if $\frac{G}{Z\left(G\right)}\cong S_4$, 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].

KEYWORDS:

• Finite group
• minimal nonabelian group
• primitive n-centraliser group

2000 MATHEMATICS SUBJECT CLASSIFICATION:

• 20D60

Acknowledgment

The authors would like to thank the referee for his/her careful reading and valuable comments.