• Refine Query
  • Source
  • Publication year
  • to
  • Language
  • 1
  • Tagged with
  • 3
  • 3
  • 3
  • 3
  • 2
  • 1
  • 1
  • 1
  • 1
  • About
  • The Global ETD Search service is a free service for researchers to find electronic theses and dissertations. This service is provided by the Networked Digital Library of Theses and Dissertations.
    Our metadata is collected from universities around the world. If you manage a university/consortium/country archive and want to be added, details can be found on the NDLTD website.
1

The lattice of normal subgroups of an infinite group

Behrendt, Gerhard Karl January 1981 (has links)
This thesis deals with various problems about the normal and subnormal structure of infinite groups. We first consider the relationship between the number of normal subgroups of a group G and of a subgroup H of finite index in G. We prove Theorem 1.5 There exists a finitely generated group G which has a subgroup H of index 2 such that H has continuously many normal subgroups and G has only countably many normal subgroups. Proposition 1.7 Let k be an infinite cardinal. Then there exists a group G of cardinality k that has only 12 normal subgroups but which contains a subgroup H of index 2 having k normal subgroups. We then consider partially ordered sets and investigate the subnormal structure of generalized wreath products. We deal with the question whether the number of subnormal subgroups of an infinite group is determined by the number of its n-step subnormal subgroups for an integer n. We prove Theorem 5.3 Let G be a group. Then G has finitely many subnormal subgroups if and only if it has finitely many 2-step subnormal subgroups. Theorem 5.5 Let m and n be infinite cardinals such that m ≤ n. Then there exists a group G with the following properties: (1) The cardinality of G is n. (2) The number of normal subgroups of G is <mathematical symbol>. (3) The number of 2-step subnormal subgroups of G is m. (4) The number of 3-step subnormal subgroups of G is 2<sup>n</sup>. Finally we consider characteristically simple groups with countably many normal subgroups. We construct a new type of characteristically simple groups: Corollary 6.15 Let ∧ be a partially ordered set such that for λ,<mathematical symbol>∊∧ there exists an automorphism a of ∧ such that <mathematical symbol> ≤ λa. Let <mathematical symbol>(∧) be the distributive lattice of semi-ideals of ∧. Then there exists a group G with the following properties: (1) |G| ≤ max(<mathematical symbol> of |<mathematical symbol>(∧)|). (2) All subnormal subgroups of G are normal in G. (3) The lattice of normal subgroups of G is isomorphic to <mathematical symbol> (∧). (4) The group G is characteristically simple.
2

On linearly ordered sets and permutation groups of uncountable degree

Ramsay, Denise January 1990 (has links)
In this thesis a set, Ω, of cardinality N<sub>K</sub> and a group acting on Ω, with N<sub>K+1</sub> orbits on the power set of Ω, is found for every infinite cardinal N<sub>K</sub>. Let W<sub>K</sub> denote the initial ordinal of cardinality N<sub>K</sub>. Define N := {α<sub>1</sub>α<sub>2</sub> . . . α<sub>n</sub>∣ 0 < n < w, α<sub>j</sub> ∈ w<sub>K</sub> for j = 1, . . .,n, α<sub>n</sub> a successor ordinal} R := {ϰ ∈ N ∣ length(ϰ) = 1 mod 2} and let these sets be ordered lexicographically. The order types of N and R are Κ-types (countable unions of scattered types) which have cardinality N<sub>K</sub> and do not embed w*<sub>1</sub>. Each interval in N or R embeds every ordinal of cardinality N<sub>K</sub> and every countable converse ordinal. N and R then embed every K-type of cardinality N<sub>K</sub> with no uncountable descending chains. Hence any such order type can be written as a countable union of wellordered types, each of order type smaller than w<sup>w</sup><sub>k</sub>. In particular, if α is an ordinal between w<sup>w</sup><sub>k</sub> and w<sub>K+1</sub>, and A is a set of order type α then A= ⋃<sub>n<w</sub>A<sub>n</sub> where each A<sub>n</sub> has order type w<sup>n</sup><sub>k</sub>. If X is a subset of N with X and N - X dense in N, then X is orderisomorphic to R, whence any dense subset of R has the same order type as R. If Y is any subset of R then R is (finitely) piece- wise order-preserving isomorphic (PWOP) to R ⋃<sup>.</sup> Y. Thus there is only one PWOP equivalence class of N<sub>K</sub>-dense K-types which have cardinality N<sub>K</sub>, and which do not embed w*<sub>1</sub>. There are N<sub>K+1</sub> PWOP equivalence classes of ordinals of cardinality N</sub>K</sub>. Hence the PWOP automorphisms of R have N<sub>K+1</sub> orbits on θ(R). The countably piece- wise orderpreserving automorphisms of R have N<sub>0</sub> orbits on R if ∣k∣ is smaller than w<sub>1</sub> and ∣k∣ if it is not smaller.
3

On irreducible, infinite, non-affine coxeter groups

Qi, Dongwen. January 2007 (has links)
Thesis (Ph. D.)--Ohio State University, 2007. / Title from first page of PDF file. Includes bibliographical references (p. 51-52).

Page generated in 0.1736 seconds