numbers measure size, groups measure symmetry. the first
statement comes as no surprise; after all, that is what numbers are
for. the second will be exploited here in an attempt to introduce
the vocabulary and some of the highlights of elementary group
theory.
a word about content and style seems appropriate. in this volume,
the emphasis is on examples throughout, with a weighting towards
the symmetry groups of solids and patterns. almost all the topics
have been chosen so as to show groups in their most natural role,
acting on （or permuting） the members ora set, whether it be the
diagonals of a cube, the edges of a tree, or even some collection
of subgroups of the given group. the material is divided into
twenty-eight short chapters, each of which introduces a new result
or idea.a glance at the contents will show that most of the
mainstays of a first course arc here. the theorems of lagrange,
cauchy, and sylow all have a chapter to themselves, as do the
classifcation of finitely generated abelian groups, the enumeration
of the finite rotation groups and the plane crystallographic
groups, and the nielsen-schreier theorem.
目錄：
preface
chapter 1 symmetries of the tetrahedron
chapter 2 axioms
chapter 3 numbers
chapter 4 dihedral groups
chapter 5 subgroups and generators
chapter 6 permutations
chapter 7 isomorphisms
chapter 8 plato''s solids and cayley''s theorem
chapter 9 matrix groups
chapter 10 products
chapter 11 lagrange''s theorem
chapter 12 partitions
chapter 13 cauehy''s theorem
chapter 14 coujugacy
chapter 15 quotient groups
chapter 16 homomorphisms
chapter 17 actions, orbits, and stabilizers
chapter 18 counting orbits
chapter 19 finite rotation groups
chapter 20 the sylow theorems
chapter 21 finitely generated abelian groups
chapter 22 row and column operations
chapter 23 automorphisms
chapter 24 the euclidean group
chapter 25 lattices and point groups
chapter 26 wallpaper patterns
chapter 27 free groups and presentations
chapter 28 trees and the nielsen-schreier theorem
bibliography
index
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