Order of groups and group elements? Let G be a group and let p be a prime. Let g and h be elements of G with order p.
I am wondering how I can use group theory to find the possible orders of the intersection between $\def\subgroup#1{\langle#1\rangle}\subgroup g$ and $\subgroup h$ and also to prove that the number of elements of order $p$ in $G$ is a multiple of $p-1$.
I've been looking for the path for ages and got nothing really. These are presented as typical applications to group theory and I'm not at ease with the subject so I'd like to see how you think on this example (in order to get a better idea). Can you hint me? Thank you.

As noted by Potato, the first thing to notice is that the intersection of two subgroups, $\langle g\rangle $ and $\langle h\rangle$ is a subgroup of $G$, but moreover, it is also a subgroup of both $\langle g\rangle$ and $\langle h\rangle$. A cyclic group of prime order, such as $\langle g\rangle$ only has two subgroups $\langle 1 \rangle$ and $\langle g\rangle$, so we see that the intersection is either trivial or the two subgroups are the same.
Consider what the above shows about subgroups of order $p$. What can their overlap look like? If you focus on $\langle g\rangle$, how many elements of order $p$ does it have? Can you use these ideas to get what you want?
