Lecture - Groups TT23, Cayley’s theorem

What theorem establishes a correspondence between actions and homomorphisms?

If you have a homomorphism $\rho : G \to \text{Sym}(S)$, how can you construct a group action (there’s no need to prove this is a group action)?

If you have a left action $\langle\cdot\rangle$ can you give an associated homomorphism (there’s no need to prove this a homomorphism)?

Can you state Cayley’s theorem?

Quickly justify that the map given by $G \to \text{Sym}(S)$ given by \(\rho (g) = (x \mapsto g \cdot x)\) is a homomorphism.

Quickly justify that there’s a valid left action given by $g \cdot x = \rho(g)(x)$ where $\rho$ is a homomorphism $\rho : G \to \text{Sym}(S)$.

Prove that, if you are given a left action of a group $G$ on $S$, then there is an associated homomorphism \(\rho : G \to \text{Sym}(S)\) and likewise for each homomorphism $\rho : G \to \text{Sym}(S)$ there is an associated left action of $G$ on $S$.

Prove (by appealing to the theorem that links left actions and homomorphisms) Cayley’s theorem:

Every finite group is isomorphic to a subgroup of a permutation group $S _ n$ for some $n$.