Primitive Subgroups of Wreath Products in Product Action

This paper is concerned with finite primitive permutation groups G which are subgroups of wreath products W in product action and are such that the socles of G and W are the same. The aim is to explore how the study of such groups may be reduced to the study of smaller groups. The O'Nan-Scott Theorem (see Liebeck, Praeger, Saxl [12] for the most recent and detailed treatment) sorts finite primitive permutation groups into several types, the groups in any one type admitting a common discussion. One of the types (III(b) in [12]) consists of groups G which are contained in, and contain the socle of, a suitable wreath product W in product action. It is easy to see that in this case the socles of G and W actually coincide. Thus the aim here amounts to pursuing the discussion of primitive groups of this type beyond the conclusions reached, say, in [12]. It has not proved possible to make direct use of those conclusions here. Instead, it seems necessary to repeat, elaborate, extend, and recombine arguments from various proofs of the O'Nan-Scott Theorem. No attempt will be made here to trace the origins of the ideas so used. For a sketch of the main results, some terminology is needed. By a wreath product A Wr Sn we mean the usual semidirect product W of the symmetric group Sn and the n-fold direct power A" of the (abstract) group A. The projection of W onto Sn corresponding to this semidirect decomposition will be denoted by n. Consider n a permutation representation of W and take a point stabilizer Wo: this has an obvious direct factorization