Order of an element in a group G Suppose that $a$ is an element of order $n$ in a group $G$. Prove:
i) $a^i = a^j$ if and only if $i \equiv j \pmod n$;
ii) if $d = (m,n)$, then the order of $a^m$ is $n/d$;
I was trying to self teach myself this and came to this question. How would you solve this? Can someone please show how to?

Hint $\rm\ (i)\ \ \ a^i = a^j\iff a^{i-j} = 1\iff n\mid i-j,\:$ since $\rm\ n = ord(a)$
$\rm(ii)\ \  \ a^{mk}\! = 1\iff n\mid mk\iff n\mid mk,nk\iff n\mid (mk,nk)=(m,n)k=dk \iff n/d\mid k$
