A Course in Group Theory

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Oxford University Press, 1996 - Business & Economics - 279 pages
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The classification of the finite simple groups is one of the major intellectual achievements of this century, but it remains almost completely unknown outside of the mathematics community. This introduction to group theory is also an attempt to make this important work better known. Emphasizing classification themes throughout, the book gives a clear and comprehensive introduction to groups and covers all topics likely to be encountered in an undergraduate course. Introductory chapters explain the concepts of group, subgroup and normal subgroup, and quotient group. The homomorphism and isomorphism theorems are explained, along with an introduction to G-sets. Subsequent chapters deal with finite abelian groups, the Jordan-Holder theorem, soluble groups, p-groups, and group extensions. The numerous worked examples and exercises in this excellent and self-contained introduction will also encourage undergraduates (and first year graduates) to further study.
 

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Contents

Definitions and examples
1
Maps and relations on sets
8
Elementary consequences of the definitions
18
Subgroups
30
Cosets and Lagranges Theorem
38
Errorcorrecting codes
49
Normal subgroups and quotient groups
59
The Homomorphism Theorem
68
Composition factors and chief factors
137
Soluble groups
146
Examples of soluble groups
155
Semidirect products and wreath products
163
Extensions
174
Central and cyclic extensions
183
Groups with at most 31 elements
192
The projective special linear groups
202

Permutations
77
The OrbitStabiliser Theorem
89
The Sylow Theorems
98
Applications of Sylow theory
106
Direct products
112
The classification of finite abelian groups
120
The JordanHolder Theorem
128
The Mathieu groups
213
The classification of finite simple groups
222
A Prerequisites from number theory and linear
234
Solutions to exercises
243
Bibliography
275
Copyright

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Symmetries
D.L. Johnson
Limited preview - 2002
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About the author (1996)

John Humphreys is at University of Liverpool.

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