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

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

An example of Order of Finite Abelian Group with $p+$ Order $p$ Elements is Divisible by $p^2$, setting $p = 3$.

$\blacksquare$

Sources

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