# Every group is a subgroup of a complete group

From Groupprops

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

## Statement

Let be a group. Then, there exists a complete group such that .

## Definitions used

`Further information: Complete group`

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

- It is centerless: its center is the trivial group.
- Every automorphism of the group is an inner automorphism.

## Related facts

### Stronger facts

- Every finite group is a subgroup of a finite simple group
- Every finite group is a subgroup of a finite complete group: The proof is the same -- we only need to observe that when the group is finite, the complete group constructed here is also finite.

## Facts used

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

## Proof

**Given**: A group .

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

**Proof**: Let . By Cayley's theorem (fact (1)), is a subgroup of . We make two cases:

- The order of is not equal to or : In this case facts (2) and (3) tell us that is a complete group, and we are done.
- The order of is equal to or : In this case, let be the symmetric group on the set , so . Further, is the symmetric group on a set of size or , which is complete, so is a subgroup of a complete group.