An Arithmetical Condition on the Sizes of Conjugacy Classes of a Finite Group

An element $x$ of a finite group $G$ is said to be primary if the order of $x$ is a prime power. We define $\mathrm{csp2}(G)$ as follows: if $|x^G|$ is a prime power for every primary element $x$ of $G$, where $x^G$ is the conjugacy class of $x$ in $G$, then $\mathrm{csp2}(G)=0$; if there exists a primary element $x$ in $G$ such that $|x^G|$ is divisible by at least two distinct primes, then $\mathrm{csp2}(G)=\max \{|x^G| | x\in G\text{is primary},|x^G|\text{is divisible by at least two distinct primes}\}$. In this paper we discuss the influence of the number $\mathrm{csp2}(G)$ on the structure of $G$.