List the left cosets $(gH)$ and right cosets $(Hg)$ for $H = \langle (123) \rangle$, where $H \le G$ and $G = S_3$.
My work so far:
$G = S_3 = \langle (12) (13) \rangle = \{ e, (12), (13), (23), (123), (132) \} $
$H = \langle (123) \rangle$
I have trouble listing out all the elements with $H$, as I'm rather iffy with generators and permutations put together. I believe this would help me tremendously on finding the cosets.