# Proving Cayley's Theorem

Im trying to understand Cayley's Theorem.I have this proof

I have some questions

What is $Sym(G)$ - I understand that it is a permutation group? How do we get that?

How do we say if $y\in G$ then $g^{-1}y\in G$?

• $\mathrm{Sym}(G)$ is the set of all permutations, that is bijections $G\rightarrow G$, together with the composition of maps. – Hagen Knaf Oct 9 '15 at 6:15
• @HagenKnaf what is composition of maps? – techno Oct 9 '15 at 6:25
• Let $f:A\rightarrow B$ and $g:B\rightarrow C$ be two maps. Then their composition $g\circ f$ is the map $A\rightarrow C$ defined by $(g\circ f)(a):=g(f(a))$. – Hagen Knaf Oct 9 '15 at 6:31
• @HagenKnaf okay... thanks – techno Oct 9 '15 at 9:14