# Group isomorphism between $D_3$ and $S_3$

If one wants to prove that $D_3$ is isomorphic to $S_3$, would it be sufficient to define a homomorphism $\psi: D_3\to S_3$ and argue that it is well-defined since $\psi(sr^i)=\psi(s)\psi(r)^i=\sigma_i\sigma_j = \psi(r^{-i}s)=\sigma_j^{-1} \sigma_i$ (where $\sigma_i$ is a transposition and $\sigma_j$ is a 3-cycle)? And then show that this homomorphism is injective and surjective.

I would show more generally that $D_n \leq S_n$. When their orders are the same, they are isomorphic. The trick is to find permutation representations of $r, s$, the generators for the Dihedral group.
• It appears that $s$ can be represented by any 2-cycle in $S_3$ and $r$ can be represented by any 3-cycle. – sequence Oct 20 '15 at 17:13