Cayley's Theorem
From AoPSWiki
Cayley's Theorem states that every group is isomorphic to a permutation group, i.e., a subgroup of a symmetric group; in other words, every group acts faithfully on some set. Although the result is simple, it is deep, as it characterizes group structure as the structure of a family of bijections.
Proof
We prove that each group 
is isomorphic to a group of bijections on itself. Indeed, for all 
, let 
be the mapping 
from into itself. Then is a bijection, for all 
; and for all 
, 
. Thus is isomorphic to the set of permutations 
on . 
The action of on itself as described in the proof is called the left action of on itself. Right action is defined similarly.
