Cayley's Theorems
John Meier · Cambridge University Press eBooks · 2008
As for everything else, so for a mathematical theory: beauty can be perceived but not explained. –Arthur Cayley An introduction to group theory often begins with a number of examples of finite groups (symmetric, alternating, dihedral, …) and constructions for combining groups into larger groups (direct products, for example). Then one encounters Cayley's Theorem, claiming that every finite group can be viewed as a subgroup of a symmetric group. This chapter begins by recalling Cayley's Theorem, then establishes notation, terminology, and background material, and concludes with the construction and elementary exploration of Cayley graphs. This is the foundation we use throughout the rest of the text where we present a series of variations on Cayley's original insight that are particularly appropriate for the study of infinite groups. Relative to the rest of the text, this chapter is gentle, and should contain material that is somewhat familiar to the reader. A reader who has not previously studied groups and encountered graphs will find the treatment presented here “brisk.” Cayley's Basic Theorem You probably already have good intuition for what it means for a group to act ona set or geometric object. For example: The cyclic group of order n – denoted ℤ n – acts by rotations on a regular n -sided polygon. The dihedral group of order 2 n – denoted D n – also acts on the regular n -sided polygon, where the elements either rotate or reflect the polygon. […]