en تخصصي Special پژوهشي Research paper <p>Let $p=(q^4+q^3+q^2+q+1)/(5,q-1)$ be a prime number, where $q$ is a prime<br> power. In this paper, we will show $Gcong mathrm{PSL}(5,q)$ if and only if<br> $|G|=|mathrm{PSL}(5,q)|$, and $G$ has a conjugacy class size $frac{|<br> mathrm{PSL}(5,q)|}{p}$. Further, the validity of a conjecture of J. G.<br> Thompson is generalized to the groups under consideration by a new way.</p> Conjugacy class size, Prime graph, Thompson's conjecture. 35 40 http://ijmsi.ir/browse.php?a_code=A-10-355-2&slc_lang=en&sid=1 A.R. Khalili Asboei khaliliasbo@yahoo.com 10031947532846006960 10031947532846006960 Yes Department of Mathematics, Farhangian University Department of Mathematics, Farhangian University