Every group is a subgroup of a complete group

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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

Let G be a group. Then, there exists a complete group H such that G \le H.

Definitions used

Further information: Complete group

A group is termed complete if it satisfies the following two conditions:

Related facts

Stronger facts

Other related facts

Facts used

  1. Cayley's theorem: Every group is a subgroup of a symmetric group -- in fact, of the symmetric group on its underlying set.
  2. Symmetric groups on finite sets are complete: The symmetric group on a finite set of size n is a complete group if n \ne 2, 6.
  3. Symmetric groups on infinite sets are complete

Proof

Given: A group G.

To prove: G is a subgroup of a complete group.

Proof: Let K = \operatorname{Sym}(G). By Cayley's theorem (fact (1)), G is a subgroup of K. We make two cases:

  • The order of G is not equal to 2 or 6: In this case facts (2) and (3) tell us that K is a complete group, and we are done.
  • The order of G is equal to 2 or 6: In this case, let H be the symmetric group on the set G \sqcup \{ x_0 \}, so G \le K \le H. Further, H is the symmetric group on a set of size 3 or 7, which is complete, so G is a subgroup of a complete group.