Book contents
- Frontmatter
- Contents
- Preface
- Twenty-five years of Groups St Andrews Conferences
- Original Introduction
- 1 An elementary introduction to coset table methods in computational group theory
- 2 Applications of cohomology to the theory of groups
- 3 Groups with exponent four
- 4 The Schur multiplier: an elementary approach
- 5 A procedure for obtaining simplified defining relations for a subgroup
- 6 GLn and the automorphism groups of free metabelian groups and polynomial rings
- 7 Isoclinisms of group extensions and the Schur multiplicator
- 8 The maximal subgroups of the Chevalley group G2(4)
- 9 Generators and relations for the cohomology ring of Janko's first group in the first twenty one dimensions
- 10 The Burnside group of exponent 5 with two generators
- 11 The orientability of subgroups of plane groups
- 12 On groups with unbounded non-archimedean elements
- 13 An algorithm for the second derived factor group
- 14 Finiteness conditions and the word problem
- 15 Growth sequences relative to subgroups
- 16 On the centres of mapping class groups of surfaces
- 17 A glance at the early history of group rings
- 18 Units of group rings: a short survey
- 19 Subgroups of small cancellation groups: a survey
- 20 On the hopficity and related properties of some two-generator groups
- 21 The isomorphism problem and units in group rings of finite groups
- 22 On one-relator groups that are free products of two free groups with cyclic amalgamation
- 23 The algebraic structure of ℵ0-categorical groups
- 24 Abstracts
- 25 Addendum to: “An elementary introduction to coset table methods in computational group theory”
- 26 Addendum to: “Applications of cohomology to the theory of groups”
- 27 Addendum to: “Groups with exponent four”
- 28 Addendum to: “The Schur multiplier: an elementary approach”
12 - On groups with unbounded non-archimedean elements
Published online by Cambridge University Press: 07 September 2010
- Frontmatter
- Contents
- Preface
- Twenty-five years of Groups St Andrews Conferences
- Original Introduction
- 1 An elementary introduction to coset table methods in computational group theory
- 2 Applications of cohomology to the theory of groups
- 3 Groups with exponent four
- 4 The Schur multiplier: an elementary approach
- 5 A procedure for obtaining simplified defining relations for a subgroup
- 6 GLn and the automorphism groups of free metabelian groups and polynomial rings
- 7 Isoclinisms of group extensions and the Schur multiplicator
- 8 The maximal subgroups of the Chevalley group G2(4)
- 9 Generators and relations for the cohomology ring of Janko's first group in the first twenty one dimensions
- 10 The Burnside group of exponent 5 with two generators
- 11 The orientability of subgroups of plane groups
- 12 On groups with unbounded non-archimedean elements
- 13 An algorithm for the second derived factor group
- 14 Finiteness conditions and the word problem
- 15 Growth sequences relative to subgroups
- 16 On the centres of mapping class groups of surfaces
- 17 A glance at the early history of group rings
- 18 Units of group rings: a short survey
- 19 Subgroups of small cancellation groups: a survey
- 20 On the hopficity and related properties of some two-generator groups
- 21 The isomorphism problem and units in group rings of finite groups
- 22 On one-relator groups that are free products of two free groups with cyclic amalgamation
- 23 The algebraic structure of ℵ0-categorical groups
- 24 Abstracts
- 25 Addendum to: “An elementary introduction to coset table methods in computational group theory”
- 26 Addendum to: “Applications of cohomology to the theory of groups”
- 27 Addendum to: “Groups with exponent four”
- 28 Addendum to: “The Schur multiplier: an elementary approach”
Summary
INTRODUCTION
Let ℓ be a length function on a group G such that N, the set of non-Archimedean elements, is a proper subset of G. Then Theorem 5.3 of shows that the lengths of elements of N are bounded if and only if ℓ(ax) = ℓ(x) for all a in N, x in G\N, in which case N is a subgroup of G and ℓ is an extension of a non-Archimedean length function on N by an Archimedean length function on G/N. Among other results, this allows the structure of any length function on an abelian group to be determined.
This leaves the question of whether, if N is a proper subgroup of G, the lengths of elements of N are necessarily bounded. In this paper we show that the answer is no. Theorem 1 of §1 gives a relation between ℓ(ax), ℓ(a) and ℓ(x), for a in N, x in G\ N, whenever N is a proper subgroup of G whose elements have unbounded lengths. This suggests that it may be possible to express such a length function in terms of a non-Archimedean length function ℓ1 = ℓ∣N on N and an Archimedean length function ℓ2 on G/N. In §2 we give an example of an HNN-group G with a length function ℓ, such that N is a proper subgroup with unbounded lengths.
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- Groups - St Andrews 1981 , pp. 228 - 236Publisher: Cambridge University PressPrint publication year: 1982
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