My book (Michael Artin's Algebra does not explain how left cosets were obtained. My difficulty understanding is from the following passage:
For example, in the symmetric group $S_{3}$, (the permutation for (123) is $x$, and the one for (12) is $y$, and so $S_{3} = \{1, x, x^{2}, y, xy, x^{2}y\}$,) the element $y$ generates a cyclic subgroup $H = \langle y\rangle$ of order 2. Then there are three left cosets of $H$ in $G$:
$ H = \{1,y\} = yH$
$ xH = \{x,xy\} = xyH$
$x^{2}H =\{x^{2},x^{2}y\} = x^{2}H$.
My question is, why are the elements in the 3 cosets grouped in the way they are?
Thanks.