The influence of order and conjugacy class length on the structure of finite groups

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

“…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).…”

“…This implies that r 1 •r 2 •r 3 •r 4 | (q 2 + q + 1) since gcd(r i , q 6 ) = gcd(r i , q 2 − q + 1) = gcd(r i , p) = 1 and r i > q + 1, which is again a contradiction. ⊲ K/H is isomorphic to 2 A 3 (2). Then p = 5, which violates the assumption p > 5.…”