Cayley's Theorem
Every group is isomorphic to a subgroup of for a suitable set , which may be finite when is finite.
Theorem 4.4 (Cayley's Theorem)
Every group is isomorphic to a subgroup of some .
Furthermore, if , we may choose with .
Proof
Take and use the left regular action. The corresponding homomorphism has trivial kernel: if , then for every , and in particular , so . Thus is an isomorphism from onto its image, a subgroup of . If is finite, then is finite.
Related
Stated in
- Theorem 4.4 (Cayley's Theorem)§4.1 Introduction