Geometry of Defining Relations in Groups: (70 Mathematics and its Applications)

The main feature of this book is a systematic application of elementary geometric and topological techniques for solving problems that arise naturally in algebra. After an account of preliminary material, there is a discussion of a geometrically intuitive interpretation of the derivation of consequences of defining relations of groups. A study is made of planar and certain other two-dimensional maps connected with well-known problems in general group theory, such as the problems of Burnside and O. Yu. Schmidt. The method of cancellation diagrams developed here is applied to these and to a series of other problems. This monograph is addressed to research workers and students in universities, and may be used as a basis for a series of specialized lectures or seminars.

Table of Contents:
1 General concepts of group theory.- §1 Definition and examples of groups.- §2 Cyclic groups and subgroups. Generators.- §3 Cosets. Factor groups. Homomorphisms.- §4 Relations in groups and free groups.- 2 Main types of groups and subgroups.- §5 p-subgroups in finite and abelian groups.- §6 Soluble groups. Laws.- §7 Finiteness conditions in groups.- 3 Elements of two-dimensional topology.- §8 Toplogical spaces.- §9 Surfaces and their cell decomposition.- §10 Topological invariants of surfaces.- 4 Diagrams over groups.- §11 Visual interpretation of the deduction of consequences of defining relations.- §12 Small cancellation theory.- §13 Graded diagrams.- 5 A-maps.- §14 Contiguity submaps.- §15 Conditions on the grading.- §16 Exterior arcs and ?-cells.- §17 Paths that are nearly geodesic and cuts on A-maps.- 6 Relations in periodic groups.- §18 Free Burnside groups of large odd exponent.- §19 Diagrams as A-maps. Properties of B(A, n).- 7 Maps with partitioned boundaries of cells.- §20 Estimating graphs for B-maps.- §21 Contiguity and weights in B-maps.- §22 Existence of ?-cells and its consequences.- §23 C-maps.- §24 Other conditions on the partition of the boundary of a map.- 8 Partitions of relators.- §25 General approach to presenting the groups G(i) and properties of these groups.- §26 Inductive step to G(i+ 1). The group G(?).- 9 Construction of groups with prescribed properties.- §27 Constructing groups with subgroups of bounded order.- §28 Groups with all subgroups cyclic.- §29 Group laws other than powers.- §30 Varieties in which all finite groups are abelian.- 10 Extensions of aspherical groups.- §31 Central extensions.- §32 Abelian extensions and dependence among relations.- 11 Presentations in free products.- §33Cancellation diagrams over free products.- §34 Presentations with condition R.- §35 Embedding theorems for groups.- §36 Operations on groups.- 12 Applications to other problems.- §37 Growth functions of groups and their presentations.- §38 On group rings of Noetherian groups.- §39 Further applications of the method.- 13 Conjugacy relations.- §40 Conjugacy cells.- §41 Finitely generated divisible groups.- Some notation.- Author Index.