Cayley's theorem

In terms of group actions
Let $$G$$ be a group. The group multiplication $$G \times G \to G$$, defines a group action of $$G$$ on itself. In other words, the left multiplication gives an action of $$G$$ on itself, with the rule $$g.h = gh$$. This action is termed the left-regular group action.

This group action is faithful -- no non-identity element of $$G$$ acts trivially.

In terms of homomorphisms
Let $$G$$ be a group. There is a homomorphism from $$G$$ to $$\operatorname{Sym}(G)$$ (the symmetric group, i.e., the group of all permutations, on the underlying set of $$G$$). Moreover, this homomorphism is injective. Thus, every group can be realized as a subgroup of a symmetric group.

In terms of group actions
Given: A group $$G$$.

To prove: $$G$$ acts on itself by left multiplication, and this gives an injective homomorphism from $$G$$ to the symmetric group on $$G$$.

Proof: Define the left-regular group action of $$G$$ on itself by $$g.h = gh$$.


 * 1) This is a group action: $$e.s = s$$ follows from the fact that $$e$$ is the identity element, while $$g.(h.s) = (gh).s$$ follows from associativity.
 * 2) The action is faithful; every non-identity element of the group gives a non-identity permutation: Assume that there are $$g, h \in G$$ such that their action by left multiplication is identical. But then $$ge = he$$ so $$g = h$$. Therefore, the action is faithful.

Thus, we get a homomorphism from $$G$$ to $$\operatorname{Sym}(G)$$. Since the action is faithful, distinct elements of $$G$$ go to distinct elements of $$\operatorname{Sym}(G)$$, so the map is injective. In particular, $$G$$ is isomorphic to a subgroup of $$\operatorname{Sym}(G)$$.

Direct applications to embedding in symmetric groups

 * Every finite group is a subgroup of a finite simple group
 * Every finite group is a subgroup of a finite complete group
 * Every group is a subgroup of a complete group
 * Finitary symmetric group on countable set is subgroup-universal for finite groups
 * Every group of given order is a permutable complement for symmetric groups

Applications to embedding in other kinds of groups

 * Every finite group is a subgroup of a linear group over any field
 * Every finite group is a subgroup of an orthogonal group over any field

Applications to embeddings for particular kinds of finite groups

 * Every group of prime power order is a subgroup of an iterated wreath product of groups of order p
 * Every group of prime power order is a subgroup of a group of unipotent upper-triangular matrices