In a cyclic group of order 60 find the elements of order 12.
then find the number of element that satisfy $x^{12}=e$ So if $x^3=e$ then $x^{12}=e$ And I know $x=e$. what next do I do?
Finally find all the elements that satisfy $x^{20}=e$. Can you make a conjecture about the number of elements that will satisfy $x^n=e$ in a cyclic group? I am completely stuck on this.
Best Answer
In general, if $g$ has order $n$, $g^k$ has order $n/(n,k)$. Can you prove this? Can you use this to solve your problem?