Date: Wed, 20 Aug 1997 22:42:05 -0400 (EDT) From: Homer BechtellProblem: For what nonabelian simple groups S does Aut(S) not split over Inn(S)? (Due to the alternating group of degree 6, the set is nonempty.) It seems reasonable that a large portion of the problem has been resolved and appears at odd places in either the literature or in theses. By a result of Bercov, if N is a minimal normal subgroup of a group G and has a nonabelian simple direct factor S such that Aut(S) splits over Inn(S), then G splits over N. If furthermore a group H has each factor in each composition series isomorphic to S (fixed), then H splits over each H-chief factor and consequently each subgroup normal in H. The hang-nail: If Aut(S) does not split over Inn(S), is the same result valid, e.g. for the alteranting group of degree 6? Recently there has been introduced a characteristic subgroup, the beta-subgroup of a group G, say b(G), such that it is of order one or it is the product of all normal subgroups N in G such that each G-chief factor in N is not complemented. The Fitting subgroup of b(G) is of course the Frattini subgroup. The only alternating group that enters as a direct factor in such a nonabelian G-chief factor is the one of degree 6. Are there other nonabelian simple groups with the same property? In the collection of primitive groups of type 2, there is the set that does not split over the socle. The automorphism group of degree 6 guarantees the set is not empty. How `full' is it? The answer to the Problem would help to resolve these questions and others of a similar nature. Obviously one can `scan' the ATLAS. But if results are already known, proper credit is demanded. Homer Bechtell