Notes - Groups TT23, Cayley’s theorem

What theorem establishes a correspondence between actions and homomorphisms?

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 $S$.

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)?

\[g \cdot s = (\rho(g))(s)\]

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

\[\rho(g) = (x \mapsto g \cdot x)\]

Can you state Cayley’s theorem?

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

Quickly justify that the map given by $G \to \text{Sym}(S)$ given by

\[\rho (g) = (x \mapsto g \cdot x)\]

is a homomorphism.

\[\rho(gh) = (x \mapsto gh \cdot x) = (y \mapsto g \cdot y) \circ (x \mapsto h \cdot x) = \rho(g) \circ \rho(h)\]

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

\[e \cdot x = \rho(e)(x) = \text{id}(x) = x\]

and

\[g \cdot (h \cdot x) = \rho(g)(h \cdot x) = \rho(g)(\rho(h)(x)) = (\rho(g) \circ \rho(h) )(x) = \rho(gh)(x) = gh \cdot x\]

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 $S$.

Todo (groups, page 83).

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 $n$.

Todo.