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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.
View a complete list of embeddability theorems

Contents 1 Statement 1.1 In terms of group actions 1.2 In terms of homomorphisms 2 Proof 2.1 In terms of group actions 3 Applications 3.1 Direct applications to embedding in symmetric groups 3.2 Applications to embedding in other kinds of groups 3.3 Applications to embeddings for particular kinds of finite groups

Statement

In terms of group actions

Let G be a group. The group multiplication , 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 (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 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 . Since the action is faithful, distinct elements of G go to distinct elements of , so the map is injective. In particular, G 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

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