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