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


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

Other related facts about complete groups

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.