Can you show why is a dihedral group homomorphic to a symmetric group?

Question

My professor mentioned something about function composition. But to show homomorphism from $G$ to $G'$, we need $f(x*y) = f(x)*f(y)$, for $x,y$ in $G$. How can you show this exactly in this case?

Answer 1

The dihedral group of order $2n$ permutes the vertices of a regular $n$-gon. Permutations form the elements of each symmetric group.

Answer 2

As far as I know, only $D_3\cong S_3$.

In general, all you can say is that a dihedral group is embeddable as a subgroup of some $S_n$. But, any group embeds into a symmetric group, by Cayley's theorem.

You may be thinking, rather, of the theorem that dihedral groups are symmetries of regular $n$-gons.