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Enumeration of Finite Groups

ISBN-10: 0521882176

ISBN-13: 9780521882170

Edition: 2007

Authors: Simon R. Blackburn, Peter M. Neumann, Geetha Venkataraman

List price: $134.99
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How many groups of order n are there? This is a natural question for anyone studying group theory, and this Tract provides an exhaustive and up-to-date account of research into this question spanning almost fifty years. The authors presuppose an undergraduate knowledge of group theory, up to and including Sylow's Theorems, a little knowledge of how a group may be presented by generators and relations, a very little representation theory from the perspective of module theory, and a very little cohomology theory - but most of the basics are expounded here and the book is more or less self-contained. Although it is principally devoted to a connected exposition of an agreeable theory, the book does also contain some material that has not hitherto been published. It is designed to be used as a graduate text but also as a handbook for established research workers in group theory.
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Book details

List price: $134.99
Copyright year: 2007
Publisher: Cambridge University Press
Publication date: 10/18/2007
Binding: Hardcover
Pages: 294
Size: 6.25" wide x 9.00" long x 0.75" tall
Weight: 1.144
Language: English

Elementary Results
Some basic observations
Groups of Prime Power Order
Tensor products and exterior squares of abelian groups
Commutators and nilpotent groups
The Frattini subgroup
Linear algebra
Enumerating p-groups: a lower bound
Relatively free groups
Proof of the lower bound
Enumerating p-groups: upper bounds
An elementary upper bound
An overview of the Sims approach
'Linearising' the problem
A small set of relations
Proof of the upper bound
Pyber's Theorem
Some more preliminaries
Hall subgroups and Sylow systems
The Fitting subgroup
Permutations and primitivity
Group extensions and cohomology
Group extensions
Restriction and transfer
The McIver and Neumann bound
Some representation theory
Semisimple algebras
Clifford's theorem
The Skolem-Noether theorem
Every finite skew field is a field
Primitive soluble linear groups
Some basic structure theory
The subgroup B
The orders of groups
Conjugacy classes of maximal soluble subgroups of symmetric groups
Enumeration of finite groups with abelian Sylow subgroups
Counting soluble A-groups: an overview
Soluble A-subgroups of the general linear group and the symmetric groups
Maximal soluble p'-A-subgroups
Enumeration of soluble A-groups
Maximal soluble linear groups
The field K and a subfield of K
The quotient G/C and the algebra <C>
The quotient B/A
The subgroup B
Structure of G determined by B
Conjugacy classes of maximal soluble subgroups of the general linear groups
Pyber's theorem: the soluble case
Extensions and soluble subgroups
Pyber's theorem
Pyber's theorem: the general case
Three theorems on group generation
Universal central extensions and covering groups
The generalised Fitting subgroup
The general case of Pyber's theorem
Other Topics
Enumeration within varieties of abelian groups
Varieties of abelian groups
Enumerating partitions
Further results on abelian groups
Enumeration within small varieties of A-groups
A minimal variety of A-groups
The join of minimal varieties
Enumeration within small varieties of p-groups
Enumerating two small varieties
The ratio of two enumeration functions
Enumerating d-generator groups
Groups with few non-abelian composition factors
Enumerating graded Lie rings
Groups of nilpotency class 3
Survey of other results
Graham Higman's PORC conjecture
Isoclinism classes of p-groups
Groups of square-free order
Groups of cube-free order
Groups of arithmetically small orders
Surjectivity of the enumeration function
Densities of certain sets of group orders
Enumerating perfect groups
Some open problems
Maximising two functions