A Family of Finite Simple Groups Which Are 2-Recognizable by Their Elements Order#

Abstract

Let G be a finite group and ω(G) the set of all orders of elements in G. Denote by h(ω(G)) the number of isomorphism classes of finite groups H satisfying ω(H) = ω(G), and put h(G) = h(ω(G)). A group G is called k-recognizable if h(G) = k < ∞ , otherwise G is called non-recognizable. In the present article we will show that the simple groups PSL(3, q), where q ≡ ±2(mod 5) and (6, (q − 1)/2) = 2, are 2-recognizable. Therefore if q is a prime power and q ≡ 17, 33, 53 or 57 (mod 60), then the groups PSL(3, q) are 2-recognizable. Hence proving the existing of an infinite families of 2-recognizable simple groups.