Subnormality and soluble factorised groups
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 th...
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ndltd-bl.uk-oai-ethos.bl.uk-2369862018-06-12T03:35:53ZSubnormality and soluble factorised groupsGold, Catharine Ann1989Throughout 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.510QA MathematicsUniversity of Warwickhttp://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.236986http://wrap.warwick.ac.uk/100929/Electronic Thesis or Dissertation |
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510 QA Mathematics Gold, Catharine Ann Subnormality and soluble factorised groups |
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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. |
author |
Gold, Catharine Ann |
author_facet |
Gold, Catharine Ann |
author_sort |
Gold, Catharine Ann |
title |
Subnormality and soluble factorised groups |
title_short |
Subnormality and soluble factorised groups |
title_full |
Subnormality and soluble factorised groups |
title_fullStr |
Subnormality and soluble factorised groups |
title_full_unstemmed |
Subnormality and soluble factorised groups |
title_sort |
subnormality and soluble factorised groups |
publisher |
University of Warwick |
publishDate |
1989 |
url |
http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.236986 |
work_keys_str_mv |
AT goldcatharineann subnormalityandsolublefactorisedgroups |
_version_ |
1718694186971037696 |