Book contents
- Frontmatter
- Contents
- Preface
- 1 Introduction to topological groups
- 2 Subgroups and quotient groups of Rn
- 3 Uniform spaces and dual groups
- 4 Introduction to the Pontryagin-van Kampen duality theorem
- 5 Duality for compact and discrete groups
- 6 The duality theorem and the principal structure theorem
- 7 Consequences of the duality theorem
- 8 Locally Euclidean and NSS-groups
- 9 Non-abelian groups
- References
- Index of terms
- Index of Exercises, propositions and theorems
8 - Locally Euclidean and NSS-groups
Published online by Cambridge University Press: 11 November 2009
- Frontmatter
- Contents
- Preface
- 1 Introduction to topological groups
- 2 Subgroups and quotient groups of Rn
- 3 Uniform spaces and dual groups
- 4 Introduction to the Pontryagin-van Kampen duality theorem
- 5 Duality for compact and discrete groups
- 6 The duality theorem and the principal structure theorem
- 7 Consequences of the duality theorem
- 8 Locally Euclidean and NSS-groups
- 9 Non-abelian groups
- References
- Index of terms
- Index of Exercises, propositions and theorems
Summary
Definition. A topological group G is said to have no small subgroups, or to be an NSS-group, if there exists a neighbourhood U of e which contains no subgroup other than {e}.
As an immediate consequence of Corollary 2 of Theorem 14 we have a complete description of compact Hausdorff abelian NSS-groups.
Proposition 45. Every compact Hausdorff abelian NSS-group is topologically isomorphic to, for some discrete group D and.
The above proposition allows us to describe all locally compact Hausdorff abelian NSS-groups.
Theorem 32. Every LCA-group G which has no small sub- groups is topologically isomorphic to, where D is some discrete group, and a and b are non-negative integers.
Proof. By the Principal Structure Theorem G has an open subgroup topologically isomorphic to, for some compact group K and. As every subgroup of an NSS-group is an NSS-group, K is an NSS-group. Proposition 45 then implies that K is topologically isomorphic to,
where S is a discrete group and. So G has an open subgroup topologically isomorphic to. Proposition 18 then shows that G is topologically isomorphic a b
to, for some discrete group.
Remark. In 1900 David Hibert presented to the International Congress of Mathematicians in Paris a series of 23 research projects (Bull. Amer. Math. Soc. 8 (1901) 437–479).
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- Publisher: Cambridge University PressPrint publication year: 1977