Past exam of the mathematics course of the University of Oxford / 2014 / Prelims / Mathematics I / 7 / a

A left group action of $G$ on a set $S$ is a function $\varphi :G\times S\to S$ satisfyingfor all $s\in S$ and $g,h\in G$. Writing $g\cdot s=\varphi (g,s)$, these are $e\cdot s=s$ and $(gh)\cdot s=g\cdot (h\cdot s)$. No commutativity is assumed, so the order of composition matters.

The group action law givesThus ${\varphi }_g$ is a bijection with inverse ${\varphi }_{g^{-1}}$, hence a permutation of $S$. Also, for every $s$,Therefore the assignment $\Phi (g)={\varphi }_g$ takes values in the symmetric group $\mathrm{Sym}(S)$ and satisfiesIt is a group homomorphism, with the group product on the right interpreted as composition of permutations.