I have a cyclic group $G$ of order $16$. A generator is $x$. I wish to find all the elements in $G$ of order $8$. I know how to find the generators and subgroups but I'm not sure how to find the elements, especially given that neither $8$ or $16$ are prime numbers..
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If $k\in\mathbb N$, then the order of $x^k$ is equal to $\frac{16}{\gcd(16,k)}$. So, the elements of order $8$ are those $x^k$ such that $\gcd(16,k)=2$.
answered Mar 31 '19 at 11:40
