Order of Finite Abelian Group with p+ Order p Elements is Divisible by p^2/Examples/Order 3/Proof 1
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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.
- $\order S \divides \order G$
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 $20$