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

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

For every finite group $G$, there exists a finite simple group $H$ containing $G$ as a subgroup. In fact, we can choose $H$ 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

Other related facts about complete groups

• Every finite group is the Fitting quotient of a finite complete group

Facts used

1. Cayley's theorem: Every finite group can be embedded in a symmetric group.
2. The symmetric group on $n$ letters can be embedded in the alternating group on $n+2$ or more letters.
3. Alternating groups are simple: The alternating group $A_{n}$ is simple non-abelian for $n\geq 5$.