Gold, Catharine Ann (1989) Subnormality and soluble factorised groups. PhD thesis, University of Warwick.

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Abstract

Throughout this summary the group G = AXB is always a product of three abelian subgroups A, X and B.

In Chapter 1 we study finite 2-groups G, where A and B are elementary and X has order 2. We also assume that X normalises both A and B, and thus AX and XB are nilpotent of class at most 2. We show that when the order of G divides 213 then G has derived length at most 3 ((1.4.2) and (1.6.1)). This supports the conjecture [see Introduction] on the derived length of a group which is expressible as the product of two nilpotent subgroups.

In Chapter 2 we consider some special cases of G where A, X and B are finite p-groups and X is cyclic. We obtain a bound for the derived length of G which is independent of the prime p and the order of X.

In Chapter 3 we find a bound for the derived length of a finite group G in terms of the highest power of a prime dividing the order of X when Ax = A, Bx = B and X is subnormal in both AX and XB. The most general result is Theorem (3.5.1). If G is a finite p-group and X has order p we show that G has derived length at most 4 (Theorem (3.3.1)).

Further in Chapter 3 if Ax « A, Bx = B, X <m AX and X <m XB then a bound for the subnormal defect of X in G is given. When X has order p this bound depends only upon m (see (3.3.4)), and when X has order pn and m is fixed then the subnormal defect of X in G can be bounded in terms of n (see the remark following Proposition (3.4.2)).

Chapter 4 shows how some results from Chapters 2 and 3 can be generalised to infinite groups. Theorem (4.3.1) shows that when A and B are p- groups of finite exponent, X has order pn, Ax = A, Bx = B, X <2 AX and X <2 XB then G is a locally finite group. Proposition (4.2.2) and Corollary (4.2.3) then enable some of the results about finite groups to be applied

Item Type:	Thesis (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Solvable groups, Finite groups, Group theory
Official Date:	9 October 1989
Dates:	Date Event 9 October 1989 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Publication Status:	Unpublished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:	University of Warwick
Format of File:	pdf
Extent:	101 leaves
Language:	eng

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