0
$\begingroup$

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..

$\endgroup$
0
$\begingroup$

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 $\gcd(16,k)=2$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.