A family of finite simple groups which are 2-recognizable by their elements order

摘要

Let G be a finite group and omega(G) the set of all orders of elements in G. Denote by h(omega(G)) the number of isomorphism classes of finite groups H satisfying omega(H) = omega(G), and put h(G) = h(omega(G)). A group G is called k-recognizable if h(G) = k < infinity, otherwise G is called non-recognizable. In the present article we will show that the simple groups PSL(3, q), where q equivalent to +/-2 (mod 5) and (6, (q - 1)/2) = 2, are 2-recognizable. Therefore if q is a prime power and q - 17, 3 3, 5 3 or 5 7 (mod 60), then the groups PSL (3, q) are 2-recognizable. Hence proving the existing of an infinite families of 2-recognizable simple groups.