Question

Prove that $D_3$ is isomorphic to $S_{\in }$.

Short answer

It is prove that $D_3$is isomorphic to $S_{\in }$.

Step-by-step solution

Define the following map on the generator of D3

$$\begin{gathered} \varnothing :D_3\to S_{\in } \\ r\to \left(123\right) \\ d\to \left(12\right) \end{gathered}$$

Define Group homomorphism

This indeed defines a group homomorphism as:

$\begin{array}{rcl}\varnothing \left(dr\right)& =& \left(12\right)\left(123\right)\\ & =& \left(132\right)\left(12\right)\\ & =& \varnothing \left(r^{-1}d\right)\end{array}$

Note now that by exercise 7.42, $S_{\in }$ is generated by $\left(123\right)$and $\left(12\right)$, which means $\varnothing$ is surjective.

Since $\left|D_3\right|=6=\left|S_{\in }\right|$, we conclude $\varnothing$ is also injective.

Therefore, an isomorphism.

Hence, $D_3$ is isomorphic to $S_{\in }$.