How can the symmetric group $S_3$ be represented as $\{1, x, x^2, y, xy, x^2y\}$

Question

On page 42 in his book Algebra, Artin begins describing the symmetric group $S_3$. He claims the six elements of the group are $\{1,x,x^2,y,xy,x^2y\}$. I don't understand how he got these elements. How did he get these 6 elements?

Answer 1

Let $x=(123), y=(23)$. That is, $x$ is a $3$ cycle and $y$ a $2$ cycle, or transposition.

Answer 2

The group $S_3$ has $6$ elements: the identity, $(1\ \ 2)$, $(1\ \ 3)$, $(2\ \ 3)$, $(1\ \ 2\ \ 3)$, and $(1\ \ 3\ \ 2)$. Now, just let $x=(1\ \ 2\ \ 3)$ and let $y=(1\ \ 2)$.