I’m having a hard time understanding cosets. I’m working through Gallians contemporary abstract and I don’t even understand example one!

Let $G=S\_3$ and $H=\{(1),(1\,3)\}$. Then it shows me the left cosets. I know $G$ is of order $6$ and $H$ is of order $2$, so there are definitely three cosets by Lagrange. 

The coset def is for ANY $a \in G$, $Ha=\{ha\mid h \in H\}$

The preliminary step is:

$(1)H,\;(12)H,\;(13)H,\;(23)H$

If $S\_3$ is $\{ 123,231,321,132,312,213\}$ why are our elements $a \in G$ above $12,13,23$. I think I get the identity one!

Further we have $1H=H$ and $13H=H$. I presume that's bc $1$ and $13$ are elements of $H$.

I totally do not get why $12H$ and $23H$ are considered and how they result in $132H$ and $123H$ respectively. I attempted the cyclic multiplication and for $12H$, I got $132H$ but with different steps and for $23H$, I got $321H$ which Ii suppose is the same as $123H$ per Gallian? But if those are identical why do we have two symmmetric members of $S\_3$, $123$ and $321$, that it seems to me are not identical?

I’m spending dozens of hours trying to understand my first abstract course, which I suppose is normal, but nonetheless incredibly frustrating. 

Thanks for your help

Marco