Order of a power of an element
Suppose
- Give a positive integer
such that . - Let
be an integer and let . Prove that the order of is .
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Suppose $G$ is a cyclic group of order $n$, and $t\in G$ is a generator.
\begin{enumerate}[label=\alph*)]
\item Give a positive integer $d$ such that $t^{-1}=t^d$.
\item Let $c$ be an integer and let $m=\gcd(n,c)$. Prove that the order of $t^c$ is $\frac{n}{m}$.
\end{enumerate}