On a Finite Group with Restriction on Set of Conjugacy Classes Size

Abstract

The greatest power of a prime p dividing the natural number n will be denoted by $n_p$. For a set of primes $\pi$ and a natural number n we will denote $n_{\pi }=\prod _{p\in \pi }n_p$. Let G be a finite group with trivial center, and $p,q>5$ be distinct prime divisors of |G|. We prove that if for every nonunity conjugacy classes size $\alpha$, it is true that $\alpha _{\{p,q\}}\in \{p^n,q^m,p^nq^m\}$, where n and m depend only on p and q, then $|G|_{\{p,q\}}=p^nq^m$, and $C_G(g)\cap C_G(h)=1$ for every p-element g and every q-element h.