Title:

Subnormality, ascendancy and projectivities

In 1939, Wielandt introduced the concept of subnormality and proved that in a finite group, the join of the two (and hence any number of) subnormal subgroups is again subnormal. This result does not hold for arbitrary groups. After much work by various authors, Williams gave necessary and sufficient conditions for the join of two subgroups to be subnormal in any group in which they are each subnormally embedded; a sufficient condition is that the two subgroups permute (i.e. their join is their product). This present work arises from considering what in some sense is the dual situation to the above, namely: given a group G with subgroups H and K , both of which contain X as a subnormal subgroup, we ask under what conditions is X subnormal in the join < H,K > of H and K? It makes sense here to assume that G = < H,K > , so we do. We will say that G is a Jgroup if whenever G = < H,K > and X are as posed, it is true that X is subnormal in G . Unfortunately, apart from obvious classes such as nilpotent groups, Jgroups do not seem to exist in abundance: Example 1.1 (due to Wielandt) shows that not even all finite groups are Jgroups. Even worse, this example has the finite group G being soluble (of derived length 3) with X central in H (in fact H 1s cyclic). All this does not seem to bode well for trying to find many infinite Jgroups (although whether metabelian groups are Jgroups is an open problem). However, Wielandt shows that, if we require that the Jgroup criteria for a group G is satisfied only when H and K permute — in which case we say that G is a ωgroup — then every finite group is indeed a ωgroup (Theorem 1.3 here). The soluble case of this result is due to Maier. Our aim in this work is to develop Theorem 1.3 in (principally) three directions, a chapter being devoted to each.
