2
$\begingroup$

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.

$\endgroup$
3
$\begingroup$

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.

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ It appears that $s$ can be represented by any 2-cycle in $S_3$ and $r$ can be represented by any 3-cycle. $\endgroup$ – sequence Oct 20 '15 at 17:13

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.