On the number of centralizers and conjugacy class sizes in finite groups

Abstract

Given a finite group G, denote by cs(G) the set of the sizes of the conjugacy classes of G and by Cent(G) the set of the centralizers of elements of G. Consider a prime p and integers $s\ge 2$ and $n\ge 2$, with gcd $(p,s)=1$. In this paper, some relations between cs(G) and $|\mathit{\text{Cent}}(G)|$ are established in the case where $\mathit{\text{cs}}(G)=\{1,p^n,p^{n-1}s\}$. Further, when $p\in \{2,3\}$, we determine the values of s and the structure of a finite group G such that $\mathit{\text{cs}}(G)=\{1,p^n,p^{n-1}s\}$. We also describe the structure of an AC-group G such that $[G:Z(G)]=3^{n}s$ and $|\mathit{\text{Cent}}(G)|=1+{\sum }_{i=0}^n{3}^i$, where gcd $(3,s)=1$.

Keywords:

• AC-group
• centralizer
• commuting graph
• conjugacy class
• Frobenius group

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 20D60
• 20E45
• 20E34
• 05C25

Acknowledgments

The author is grateful to Professor Irene N. Nakaoka for useful comments during this writing. He also thanks the anonymous referee for careful reading of the article.

Disclosure statement

The author has no competing interests to declare that are relevant to the content of this article.

Additional information

Funding

No funding was received to assist with the preparation of this manuscript.