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

For G a finite group, π^sub ^sub e^^(G) denotes the set of orders of elements in G. If Ω is a subset of the set of natural numbers, h(Ω) stands for the number of isomorphism classes of finite groups with the same set Ω of element orders. We say that G is k-distinguishable if h(π^sub ^sub 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^sub 12^, M^sub 22^, J^sub 2^, He, Suz, M^sup ^sup 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^sub 12^, M^sub 22^, He, Suz or O'N, then h(π^sub ^sub e^^(Aut(M))) ¸ {1,∞}.[PUBLICATION ABSTRACT]