Finite Groups With Product Fusion

Abstract

In several recent papers concerning the classification of finite simple groups, the investigators were forced to treat subsidiary problems involving a group G whose Sylow 2-subgroup S is a direct product S = S, x S2 in which the fusion of 2-elements of G corresponds to that of the direct product of two groups having S, and S2, respectively, as their Sylow 2-subgroups. For example, in the study of groups, with Sylow 2-subgroups of type G2(q) or Psp(4, q), q odd, the determination of the structure of the centralizer of a central involution requires a solution of this subsidiary problem in the case in which S, and S2 are dihedral groups ([9]). The analogous problem arises with S, dihedral and S2 wreathed in Mason's work on groups with Sylow 2-subgroups of type L4(q), q odd. Likewise in the analysis of groups with Sylow 2-subgroups of type Psp(6, q), q odd, two other such direct product problems arise in the course of determining the structure of the centralizers of involutions. It would therefore appear to be quite useful to have available an effective result about the structure of such a group G when the direct factors S, S2 of S are arbitrary. It is the object of this paper to establish such a general result. Suppose that we are trying to determine all simple groups G satisfying some set of conditions. In such a problem, it is very likely that one can reduce to the case in which the critical composition factors of the proper subgroups of G are of known type. Moreover, in those cases in which such a classification theorem has been obtained, specific properties of such composition factors have entered into the arguments. It is therefore reasonable and most likely necessary to impose some general conditions on appropriate composition factors of the proper subgroups of the group G under investigation. It turns out that a single assumption stated in terms of the notion of balance is all that is necessary. Moreover, we make this assumption only on the composition factors of certain subgroups of G which possess a Sylow 2-subgroup of the form T1 x T2with T, z Si, i = 1, 2.