Symmetry groups in algebra

Question

Recently I was going over online notes regarding symmetry groups and I came across the following notation:

$S_3=\{1,x,x^2,y,xy,x^2y\}$ is generated by $\{x,y\}$. What does this mean? Aren't the elements in $S_3$ of the form $\{(12),(123),(23),(132),e, (13)\}$. Can someone please explain?

Answer 1

Take $x=(123)$ and $y=(12)$. Note that $x^3=e$ and $y^2=e$.