When we began to consider the scope of this book, we envisaged a catalogue supplying at least one abstract definition for any finitely-generated group that the reader might propose. But we soon realized that more or less arbitrary restrictions are necessary, because interesting groups are so numerous. For permutation groups of degree 8 or less (i.e., subgroups of S_{8}), the reader cannot do better than consult the tables of JOSEPHINE BURNS (1915), while keeping an eye open for misprints. Our own tables (on pages 134-142) deal with groups of low order, finite and infinite groups of congruent transformations, symmetric and alternating groups, linear fractional groups, and groups generated by reflections in real Euclidean space of any number of dimensions.
The best substitute for a more extensive catalogue is the description (in Chapter 2) of a method whereby the reader can easily work out his own abstract definition for almost any given finite group. This method is sufficiently mechanical for the use of an electronic computer.
There is also a topological method (Chapter 3), suitable not only for groups of low order but also for some infinite groups. This involves choosing a set of generators, constructing a certain graph (the Cayley diagram or DEHNsche Gruppenbild), and embedding the graph into a surface. Cases in which the surface is a sphere or a plane are described in Chapter 4, where we obtain algebraically, and verify topologically, an abstract definition for each of the 17 space groups of two-dimensional crystallography.
In Chapter 5, the fundamental groups of multiply-connected surfaces are exhibited as symmetry groups in the hyperbolic plane, the generators being translations or glide-reflect ions according as the surface is orientable or non-orientable.
The next two chapters deal with special groups that have become famous for various reasons. In particular, certain generalizations of the polyhedral groups, scattered among the numerous papers of G A MILLER, are derived as members of a single family. The inclusion of a slightly different generalization in § 6.7 is justified by its unexpected connection with SHEPHARD's regular complex polygons.
Chapter 8 pursues BRAHANA'S idea that any group generated by two elements, one of period 2, can be represented by a regular map or topological polyhedron.
In Chapter 9 we prove that every finite group defined by relations of the form
(R_{j)2}= (R_{j}R_{k})^{pjk} = E (1 ≤ j < k ≤ n)
can be represented in Euclidean n-space as a group generated by reflections in n hyperplanes. Many well-known groups belong to this family. Some of them play an essential role in the theory of simple Lie groups.
We wish to express our gratitude to Professor REINHOLD BAER for inviting us to undertake this work and for constructively criticizing certain parts of the manuscript. In the latter capacity we would extend our thanks also to Dr PATRICK Du VAL, Professor IRVINE REINER, Professor G de B ROBINSON, Dr F A SHERK, Dr J A TODD and Professor A W TUCKER. We thank Mr J F PETRIE for two of the drawings: Figs. 4.2, 4.3; and we gratefully acknowledge the assistance of Mrs BERYL MOSER in preparing the typescript.
H. S. M. C.
University of Toronto
W. 0. J. M.
University of Saskatchewan
February 1957
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