On Normal Subgroups Which Are Direct Products

Now that the classification of finite simple groups is complete, it is logical to look at the extension problem. An important special case to consider is when M is a minimal normal subgroup of G and both G/M and M are known groups. If M is abelian, various techniques have been used to derive information about G. Indeed, almost the entire theory of finite solvable groups can be said to rest upon these techniques. The motivation behind the present paper was to develop techniques for dealing with the situation when M is not abelian. Specifically, we consider the following problems: (1) Determine the structure of G from the structure of G/M and some subgroup or subgroups of G. (2) Find subgroups H in G such that G = HM, and, in particular, determine whether M has a complement in G. (3) Determine when two subgroups H, and H, found in (2) are conjugate in G. If M is a non-abelian minimal normal subgroup of a finite group G, then