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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
- 1 Statement
- 2 Proof
- 3 Applications
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.
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 .
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