How should I write down the alternating group $A_3$? I didn't understand how to write down the alternating group A3. 
Is this the group consisting of only the even permutations? Also, what familiar group is this isomorphic to?

You can also have an isomorphism with the following matrix group:
$$I=\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}, \ \ C=\begin{pmatrix}0&0&1\\1&0&0\\0&1&0\end{pmatrix}, \ \ C^2=\begin{pmatrix}0&1&0\\0&0&1\\1&0&0\end{pmatrix}$$
Explanation: if $e_1,e_2,e_3$ is the canonical basis


*

*$I$ will sent all $e_i$ on themselves, 

*$C$ will send $e_1$ onto $e_2$,  $e_2$ onto $e_3$,  $e_3$ onto $e_1$, that's a cycle !  

*$C^2$ will... be as well $C^{-1}$, because $C^3=I.$
