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Generators and Relations for Discrete Groups (Paperback, 1957)
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Generators and Relations for Discrete Groups (Paperback, 1957)
Series: Ergebnisse der Mathematik und Ihrer Grenzgebiete. 1. Folge, 14
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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 e ), the reader cannot do
better than consult the 8 tables of JosEPHINE BuRNS (1915), while
keeping an eye open for misprints. Our own tables (on pages
134-143) deal with groups of low order, finiteandinfinite 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
foramoreextensive 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 DEHNsehe 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."
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