Order of Finite Abelian Group with p+ Order p Elements is Divisible by p^2/Examples/Order 3/Proof 1

Proof
By hypothesis there are elements $x, y$ of order $3$ in $G$ such that $x, y, x^2$ are all different.

Consider the subset of $G$:
 * $S := \set {x^i y^j: 0 \le i, j \le 2}$

By the Finite Subgroup Test, $S$ is a subgroup of $G$ which has $9$ elements.

By Lagrange's theorem:
 * $\order S \divides \order G$

But $\order S = 9$ and so $9 \divides \order G$.