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

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 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 $$n \ge 5$$.