Automorphism groups of sporadic groups

Abstract: Among the simplest invariants of the sporadic finite simple groups are their outer automorphism groups. For 12 of the 26 possible isomorphism types of a sporadic simple group G, the outer automorphism group Out(G) has order 2, and in the remaining 14 cases, Out(G) is trivial. Historically the suspicion of the existence of a sporadic group was followed in fairly short order by the calculation of a good upper bound on the size of its outer automorphism group. In a few cases establishing the existence of certain outer automorphisms, like the existence of the groups themselves, presented difficulties overcome only with the use of machine computation. In any case these calculations can be difficult to track down in the literature. This note, which contains nothing new, is only intended to bring together these calculations. It also answers a question of Bob Oliver about the automorphism groups of some of these groups, how they might be calculated, and specifically whether the Sylow 2-subgroups of a sporadic simple group are self-centralizing in the automorphism group of the simple group. The answer is that they are.