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 and , with gcd . In this paper, some relations between cs(G) and are established in the case where . Further, when , we determine the values of s and the structure of a finite group G such that . We also describe the structure of an AC-group G such that and , where gcd .
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.