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

For G a finite group,π(G)denotes the set of orders of elements in G.If Ω is a subsetof the set of natural numbers,h(Ω)stands for the number of isomorphism classes of finite groups withthe same set Ω of element orders.We say that G is k-distinguishable if h(π)=k<∞,otherwiseG is called non-distinguishable.Usually,a 1-distinguishable group is called a characterizable group.Itis shown that if M is a sporadic simple group different from M,M,J,He,Suz,M~cL and O’N,then Aut(M)is charaeterizable by its dement orders.It is also proved that if M is isomorphic toM,M,He,Suz or O’N,then h(π(Aut(M)))∈{1,∞}.