$G$ is a finite group, and the set $X=\{a\in G: o(a)=n\}$ where $n$ is a positive integer.
I defined the relation on $X$ as:
$a\mathrel{R}b$ if $\langle a\rangle=\langle b\rangle$ i.e. the cyclic group generated by $a$ is equal to the one generated by $b$.
It was fine for me to say it is reflexive as $a\mathrel{R}a$ implies $\langle a\rangle=\langle a\rangle$, and symmetric because $\langle a\rangle=\langle b\rangle$ implies $\langle b\rangle=\langle a\rangle$. But now I got stuck for proving the transitivity of this relation: is it really simple to say if $\langle a\rangle=\langle b\rangle$ and $\langle b\rangle=\langle c\rangle$, so $\langle a\rangle=\langle c\rangle$ thus $a\mathrel{R}c$?
In addition, I feel a little bit tough to show how to show that the number of $a$ in $G$ whose order is $n$ can be written as a multiple of $\phi(n)$, where $\phi$ refers to the Euler's $\phi$ function. Can I be helped to do a quick proof?