# Cayley's theorem

This article describes a fact or result that is not basic but it still well-established and standard. The fact may involve terms that are themselves non-basic

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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.

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## Contents

## Statement

### In terms of group actions

Let be a group. The group multiplication , defines a group action of on itself. In other words, the *left multiplication* gives an action of on itself, with the rule . This action is termed the left-regular group action.

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

### In terms of homomorphisms

Let be a group. There is a homomorphism from to (the symmetric group, i.e., the group of all permutations, on the underlying set of ). 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 .

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

**Proof**: Define the left-regular group action of on itself by .

- This is a group action: follows from the fact that is the identity element, while follows from associativity.
- The action is faithful; every non-identity element of the group gives a non-identity permutation: Assume that there are such that their action by left multiplication is identical. But then so . Therefore, the action is faithful.

Thus, we get a homomorphism from to . Since the action is faithful, distinct elements of go to distinct elements of , so the map is injective. In particular, is isomorphic to a subgroup of .

## 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