Allen, Peter S. (1987) Subnormality, ascendancy and projectivities. PhD thesis, University of Warwick.

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Abstract

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 J-group 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, J-groups do not seem to exist in abundance: Example 1.1 (due to Wielandt) shows that not even all finite groups are J-groups. 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 J-groups (although whether metabelian groups are J-groups is an open problem). However, Wielandt shows that, if we require that the J-group criteria for a group G is satisfied only when H and K permute — in which case we say that G is a ω-group — then every finite group is indeed a ω-group (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.

Item Type:	Thesis (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Finite groups, Isomorphisms (Mathematics), Lattice theory
Official Date:	October 1987
Dates:	Date Event October 1987 Submitted
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Stonehewer, Stewart E. (Stewart Edward),‎ 1935-
Sponsors:	Science and Engineering Research Council (Great Britain)
Format of File:	pdf
Extent:	iv, 125 leaves : illustrations
Language:	eng

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