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

Example of Order of Finite Abelian Group with $p+$ Order $p$ Elements is Divisible by $p^2$

Let $G$ be a finite abelian group whose identity is $e$.

Let $G$ have more than $2$ elements of order $3$.

Then:

$9 \divides \order G$

where:

$\divides$ denotes divisibility

$\order G$ denotes the order of $G$.

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 $9 \divides \order G$.

$\blacksquare$

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $7$: Cosets and Lagrange's Theorem: Exercise $20$