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

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Official URL: http://webcat.warwick.ac.uk/record=b3158003~S15
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 2groups 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 pgroups 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 pgroup 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: 


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Publication Status:  Unpublished  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:  
Extent:  101 leaves  
Language:  eng 
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