Exercise 1.1.23 (If $|x| = st$ then $|x^s| = t$)

Question

Suppose $x \in G$ and $|x| = n < \infty$. If $n = st$ for some positive integers $s$ and $t$, prove that $|x^s| = t$.

Richard Ganaye · Accepted Answer

ewcommand{\Z}{\mathbb{Z}}

\begin{proof} 
Suppose $x \in G$ and $|x| = n < \infty$, where $n = st,\ s>0, \ t>0$.
For all $k$ in $\Z$, 
$$(x^s)^k = 1 \iff x^{sk} = 1 \iff n \mid sk \iff st \mid sk \iff t \mid k. $$
Then the least positive integer $k$ such that $(x^s)^k = 1$ is $t$, so
$$|x^s| = t.$$
\end{proof}

Exercise 1.1.23 (If $|x| = st$ then $|x^s| = t$)

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Exercise 1.1.23 (If $|x| = st$ then $|x^s| = t$)

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