# Every finite group is a subgroup of a finite simple non-abelian 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

For every finite group , there exists a finite simple group containing as a subgroup. In fact, we can choose to be a finite simple non-abelian group.

## Related facts

### Related facts about embedding as subgroups

- Every finite group is a subgroup of a finite complete group
- Every group is a subgroup of a complete group
- Every aperiodic group is a subgroup of a simple aperiodic group
- Every aperiodic group is a subgroup of an aperiodic group with two conjugacy classes

## Facts used

- Cayley's theorem: Every finite group can be embedded in a symmetric group.
- The symmetric group on letters can be embedded in the alternating group on or more letters.
- Alternating groups are simple: The alternating group is simple non-abelian for .