Geometry of Defining Relations in Groups [Paperback]

1 General concepts of group theory.- ?1 Definition and examples of groups.- 1 Definition of group.- 2 Examples of groups.- 3 Group isomorphism.- ?2 Cyclic groups and subgroups. Generators.- 1 Subgroups.- 2 Cyclic groups.- 3 Subgroups of cyclic groups.- 4 Sets of generators.- ?3 Cosets. Factor groups. Homomorphisms.- 1 Decomposition of a group into cosets.- 2 Normal subgroups and factor groups.- 3 Homomorphism theorems.- ?4 Relations in groups and free groups.- 1 Free groups.- 2 Defining relations.- 3 Words and subwords.- 2 Main types of groups and subgroups.- ?5 p-subgroups in finite and abelian groups.- 1 Conjugacy class. The centre.- 2 p-subgroups of finite groups.- 3 Direct products.- 4 Primary decompositions of abelian groups.- ?6 Soluble groups. Laws.- 1 The derived group.- 2 Soluble groups.- 3 Soluble and finite simple groups.- 4 Laws and varieties.- ?7 Finiteness conditions in groups.- 1 Local finiteness. The conditions max and min.- 2 Soluble Noetherian and Artinian groups.- 3 The role of involutions.- 3 Elements of two-dimensional topology.- ?8 Toplogical spaces.- 1 The definitions of topological and metric spaces.- 2 Continuous mappings.- 3 Quotient spaces.- 4 Compactness.- 5 Connectedness.- ?9 Surfaces and their cell decomposition.- 1 The Jordan curve theorem.- 2 The combinatorial definition of a surface.- 3 Comparison of triangulations.- 4 Cell decompositions of surfaces.- 5 Graphs on a surface.- ?10 Topological invariants of surfaces.- 1 The Euler characteristic.- 2 Consequences for graphs.- 3 Orientable surfaces.- 4 The fundamental group of a cell decomposition.- 5 Computation of the fundamental groups of surfaces.- 4 Diagrams over groups.- ?11 Visual interpretation of the deduction of consequences of defining relations.- 1 Some examples.- 2 The concept of a diagram.- 3 von Kampens lemma.- 4 Annular diagrams; subdiagrams.- 5 0-refinements of diagrams.- 6 Cancellable pairs of cells.- ?12 Small cancellation theory.- 1 The conditions C(?) and C(k).- 2 Dl#|