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.