Characterization of $mathrm{PSL}(5,q)$ by its Order and One Conjugacy Class Size

Abstract

Let p = q 4 +q 3 +q 2 +q+1 (5,q−1) be a prime number, where q is a prime power. In this paper, we will show G ∼ = PSL(5, q) if and only if |G| = |PSL(5, q)|, and G has a conjugacy class size |PSL(5,q)| p . Further, the validity of a conjecture of J. G. Thompson is generalized to the groups under consideration by a new way.