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”
9 - Generators and relations for the cohomology ring of Janko's first group in the first twenty one dimensions
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
If G is a finite group, and A a G-ring, then generators and relations for H*(G,A), the cohomology of G with coefficients in A, may be obtained via the Hochschild-Serre spectral sequence. Examples include. Prerequisite to such a calculation is the existence of a proper normal subgroup of G, so that for simple groups this method is not available. Perhaps, in such cases, a more appropriate method is to extract the p-part of the cohomology ([H*(G,A)]p) from the cohomology ring of the Sylow subgroup for each prime p dividing ∣G∣, the order of G ([1, p.259]). In general, this requires a knowledge of the intersection of the Sylow p-subgroups, but has the following two simplifications. Let Gp be the Sylow p-subgroup of G, and Φp the group of automorphisms of Gp induced by inner automorphisms of G.
Lemma 1 ([12, Lemma 1]). If Gpis abelian, then [H*(G,A)] consists of those elements of H*(Gp,A) which are fixed under the action of Φpon H*(Gp,A).
Lemma 2 ([12, Theorem 2]). If p is odd and Gpis cyclic, then the p-period of G is twice the order of Φp.
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- Groups - St Andrews 1981 , pp. 201 - 206Publisher: Cambridge University PressPrint publication year: 1982
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