New characterization of symmetric groups of prime degree

Abstract

Abstract. In this paper, will show that a symmetric group of prime degree p ≥ 5 is recognizable by its order and a special conjugacy class size of (p − 1)!.

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

“…Similar characterizations have been found in [5], [2] , [8], [6], [7] and [5] for the groups: PSL(n, 2), 2 D n (2), 2 D n+1 (2), C n (2) , alternating group of degree p, p + 1, p + 2 and symmetric group of degree p, where p is a prime number.…”

A variation of Thompson's conjecture for the symmetric groups

“…Asboei and Mohammadyari used the group order and just one special conjugacy class size to verify Thompson's conjecture for alternating simple groups of degrees p, p + 1 and p + 2, where p is a prime number, see [3] and [4]. They also extended their result to symmetric groups of prime degrees, see [5]. Furthermore, Asboei et al in [2] recently showed that simple symplectic groups PSp 2n (2) are determined uniquely up to isomorphism by their order and one conjugacy class of size |PSp 2n (2)|/(2 n + 1).…”