Finitary symmetric group on countable set is subgroup-universal for finite groups
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
- Free group on countable set is quotient-universal for finitely generated groups
- Free group on two generators is SQ-universal
- Cayley's theorem: This states that every group embeds as a subgroup of the symmetric group on its underlying set.
The proof follows from fact (1), and the observation that the symmetric group on any finite set is isomorphic to some subgroup of -- namely, the subgroup comprising the permutations on a finite subset of of the same cardinality.