Order of Group Element/Examples/Possible Orders of x when x^2 = x^12
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Examples of Order of Group Element
Let $G$ be a group.
Let $x \in G \setminus \set e$ be such that $x^2 = x^{12}$.
Then the possible orders of $x$ are $2$, $5$ and $10$.
Proof
\(\ds x^2\) | \(=\) | \(\ds x^{12}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds x^2\) | \(=\) | \(\ds x^2 \cdot x^{10}\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds x^{10}\) | \(=\) | \(\ds e\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \order x\) | \(\divides\) | \(\ds 10\) |
Hence:
- $\order x \in \set {2, 5, 10}$
as required.
$\blacksquare$
Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $6$: An Introduction to Groups: Exercise $11$