Cayley's Representation Theorem/Proof 1

Theorem

Let $S_n$ denote the symmetric group on $n$ letters.

Every finite group is isomorphic to a subgroup of $S_n$ for some $n \in \Z$.

Proof

Let $H = \set e$.

We can apply Permutation of Cosets to $H$ so that:

$\mathbb S = G$

and:

$\map \ker \theta = \set e$

The result follows by the First Isomorphism Theorem for Groups.

$\blacksquare$

Source of Name

This entry was named for Arthur Cayley.

Sources

• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $9$: Permutations: Corollary $9.24$