Cayley's theorem

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This article gives the statement, and possibly proof, of an embeddability theorem: a result that states that any group of a certain kind can be embedded in a group of a more restricted kind.
View a complete list of embeddability theorems

Statement

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.

Proof

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 $\operatorname {Sym} (G)$.

Applications

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