Let $p$ be prime, and let $Z=\{z\in \Bbb C: z^{p^n}=1 \text{ for some } n\in\Bbb Z^+\}$.
I see that $Z$ is indeed group, but, I wonder if we can replace '$p$ is prime' with '$p$ any positive integer'?
I proved $Z$ is indeed a group by showing that it is subgroup of group of roots of unity: $Z$ indeed contains identity, if $x,y$ are in $Z$, then $x^{p^k}=1$ and $y^{p^l}=1$ for some $k,l\in\Bbb Z^+$ $(xy)^{\text{lcm} (p^k,p^l)}=1$ and $x^{-1}$ lies indeed in $Z$. I thought that it was ${\text{lcm} (p^k,p^l)}=1$ where '$p$ is prime' was needed, to see that ${\text{lcm} (p^k,p^l)}$ is indeed positive power of $p$. But I was wrong.
If taking $p$ is positive integer works, then why do we cherish the '$p$ prime' version