# A Course in the Theory of Groups

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September 1993

## Beschreibung

### Beschreibung

" A group is defined by means of the laws of combinations of its symbols," according to a celebrated dictum of Cayley. And this is probably still as good a one-line explanation as any. The concept of a group is surely one of the central ideas of mathematics. Certainly there are a few branches of that science in which groups are not employed implicitly or explicitly. Nor is the use of groups confined to pure mathematics. Quantum theory, molecular and atomic structure, and crystallography are just a few of the areas of science in which the idea of a group as a measure of symmetry has played an important part. The theory of groups is the oldest branch of modern algebra. Its origins are to be found in the work of Joseph Louis Lagrange (1736-1813), Paulo Ruffini (1765-1822), and Evariste Galois (1811-1832) on the theory of algebraic equations. Their groups consisted of permutations of the variables or of the roots of polynomials, and indeed for much of the nineteenth century all groups were finite permutation groups. Nevertheless many of the fundamental ideas of group theory were introduced by these early workers and their successors, Augustin Louis Cauchy (1789-1857), Ludwig Sylow (1832-1918), Camille Jordan (1838-1922) among others. The concept of an abstract group is clearly recognizable in the work of Arthur Cayley (1821-1895) but it did not really win widespread acceptance until Walther von Dyck (1856-1934) introduced presentations of groups.### Inhaltsverzeichnis

1 Fundamental Concepts of Group Theory.- 1.1 Binary Operations, Semigroups, and Groups.

- 1.2 Examples of Groups.

- 1.3 Subgroups and Cosets.

- 1.4 Homomorphisms and Quotient Groups.

- 1.5 Endomorphisms and Automorphisms.

- 1.6 Permutation Groups and Group Actions.- 2 Free Groups and Presentations.

- 2.1 Free Groups.

- 2.2 Presentations of Groups.

- 2.3 Varieties of Groups.- 3 Decompositions of a Group.

- 3.1 Series and Composition Series.

- 3.2 Some Simple Groups.

- 3.3 Direct Decompositions.- 4 Abelian Groups.

- 4.1 Torsion Groups and Divisible Groups.

- 4.2 Direct Sums of Cyclic and Quasicyclic Groups.

- 4.3 Pure Subgroups and p-groups.

- 4.4 Torsion-free Groups.- 5 Soluble and Nilpotent Groups.

- 5.1 Abelian and Central Series.

- 5.2 Nilpotent Groups.

- 5.3 Groups of Prime-Power Order.

- 5.4 Soluble Groups.- 6 Free Groups and Free Products.

- 6.1 Further Properties of Free Groups.

- 6.2 Free Products of Groups.

- 6.3 Subgroups of Free Products.

- 6.4 Generalized Free Products.- 7 Finite Permutation Groups.

- 7.1 Multiple Transitivity.

- 7.2 Primitive Permutation Groups.

- 7.3 Classification of Sharply k-transitive Permutation Groups.

- 7.4 The Mathieu Groups.- 8 Representations of Groups.

- 8.1 Representations and Modules.

- 8.2 Structure of the Group Algebra.

- 8.3 Characters.

- 8.4 Tensor Products and Representations.

- 8.5 Applications to Finite Groups.- 9 Finite Soluble Groups.

- 9.1 Hall ?-subgroups.

- 9.2 Sylow Systems and System Normalizers.

- 9.3 p-soluble Groups.

- 9.4 Supersoluble Groups.

- 9.5 Formations.- 10 The Transfer and Its Applications.

- 10.1 The Transfer Homomorphism.

- 10.2 Grün's Theorems.

- 10.3 Frobenius' Criterion for p-nilpotence.

- 10.4 Thompson's Criterion for p-nilpotene.

- 10.5 Fixed-point-free Automorphisms.- 11 The Theory of Group Extensions.

- 11.1 Group Extensions and Covering Groups.

- 11.2 Homology Groups and Cohomology Groups.

- 11.3 The Gruenberg Resolution.

- 11.4 Group-theoretic Interpretations of the (Co)homology Groups.- 12 Generalizations of Nilpotent and Soluble Groups.

- 12.1 Locally Nilpotent Groups.

- 12.2 Some Special Types of Locally Nilpotent Groups.

- 12.3 Engel Elements and Engel Groups.

- 12.4 Classes of Groups Defined by General Series.

- 12.5 Locally Soluble Groups.- 13 Subnormal Subgroups.

- 13.1 Joins and Intersections of Subnormal Subgroups.

- 13.2 Permutability and Subnormality.

- 13.3 The Minimal Condition on Subnormal Subgroups.

- 13.4 Groups in Which Normality Is a Transitive Relation.

- 13.5 Automorphism Towers and Complete Groups.- 14 Finiteness Properties.

- 14.1 Finitely Generated Groups and Finitely Presented Groups.

- 14.2 Torsion Groups and the Burnside Problems.

- 14.3 Locally Finite Groups.

- 14.4 2-groups with the Maximal or Minimal Condition.

- 14.5 Finiteness Properties of Conjugates and Commutators.- 15 Infinite Soluble Groups.

- 15.1 Soluble Linear Groups.

- 15.2 Soluble Groups with Finiteness Conditions on Abelian Subgroups.

- 15.3 Finitely Generated Soluble Groups and the Maximal Condition on Normal Subgroups.

- 15.4 Finitely Generated Soluble Groups and Residual Finiteness.

- 15.5 Finitely Generated Soluble Groups and Their Frattini Subgroups.

EAN: 9780387940922

ISBN: 0387940928

Untertitel: Softcover Repri.
Sprache: Englisch.

Verlag: SPRINGER VERLAG GMBH

Erscheinungsdatum: September 1993

Seitenanzahl: 504 Seiten

Format: kartoniert

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