A RECOGNITION OF SIMPLE GROUPS PSL(3, q) BY THEIR ELEMENT ORDERS

For any group G, denote by πe(G) the set of orders of elements in G. Given a finite group G, let h(πe(G)) be the number of isomorphism classes of finite groups with the same set πe(G) of element orders. A group G is called k-recognizable if h(πe(G))= k < ∞, otherwise G is called non-recognizable. Also a 1-recognizable group is called a recognizable (or characterizable) group. In this paper the authors show that the simple groups PSL(3, q), where 3 < q ≡ ±2 (mod 5) and (6, (q – 1)/2) = 1, are recognizable.