On the Characterizability of the Automorphism Groups of Sporadic Simple Groups by Their Element Orders

For G a finite group, e (G) denotes the set of orders of elements in G. If is a subset of the set of natural numbers, h() stands for the number of isomorphism classes of finite groups with the same set of element orders. We say that G is k-distinguishable if h( e (G)) = k > , otherwise G is called non-distinguishable. Usually, a 1-distinguishable group is called a characterizable group. It is shown that if M is a sporadic simple group different from M 12, M 22, J 2, He, Suz, M c L and O'N, then Aut(M) is characterizable by its element orders. It is also proved that if M is isomorphic to M 12, M 22, He, Suz or O'N, then h( e (Aut(M))) {1,infinity}.