Let $G$ be a group of order $n$ then $G$ is isomorphic to a subgroup of $S_n$,Denote it by $\phi$, $\phi:G\to S_n$ be an monomorphism of Cayleys Theorem
Let $g\in G$ has order $k$.Show that $\phi(g)$ is a product of disjoint cycles of length $k$.Moreovver Show that $\phi(g)$ is an even permutation unless $g$ has even order and $\langle g\rangle $ has odd index in $G$.
Since $g$ has order $k$ so $\phi(g)$ also has order $k$ and hence is either a $k-$ cycle or a product of $k$ cycles.
Given that $g$ has even order and so $\phi(g)$ has even order and hence it is a product of odd number of transpositions.
But I need to prove that it is even permutation?
Where am I missing the point?