Cayley's theorem

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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.
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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

Applications to embedding in other kinds of groups

Applications to embeddings for particular kinds of finite groups