Date: Wed, 20 Aug 1997 22:42:05 -0400 (EDT)
From: Homer Bechtell 

Problem: 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