Number of elements of order $p$ in a (possibly infinite) group If a group is finite, then it's easy to see by group action that $n_p\equiv-1\pmod p$.
However, is the result true for an infinite group? My conjecture is that if $n_p$ is non-zero and finite, then $n_p\equiv-1\pmod p$ of infinite groups. I proved the result for $n_2$, but don't know whether it is true for all primes.
 A: We know that the result is true for finite groups (by the group-action argument you're familiar with).
But if there's only finitely many elements of order $p$, we can look at $H$ the group generated by those elements. It is finite, and contains all the elements of order $p$. So the result holds for $H$, and since $H$ has all the elements of order $p$, the result holds for $G$.

I hope this helps ^_^
Added by Derek Holt: proof that $H$ is finite. Since $H$ is generated by the elements of order $p$ and there are only finitely many, they have only finitely many conjugates, so their centralizers have finite index in $H$. So the intersection of their centralizers, which is the centre $Z(H)$ of $H$ has finite index in $H$.
Now, by a result of Schur, the derived group $[H,H]$ is finite. Since $H/[H,H]$ is an abelian group generated by finitely many elements of order $p$, it is also finite, and hence $H$ is finite.
A: Suppose $G$ is a group with finitely many elements of order $p$.  We may assume that $G$ is generated by its elements of order $p$ (otherwise, just look at the subgroup they generate).  Since every conjugate of an element of order $p$ has order $p$, each element of order $p$ must have centralizer of finite index.  So, the intersection $Z$ of the centralizers of elements of order $p$ is finite index.  Since $G$ is generated by its elements of order $p$, $Z$ is actually the center of $G$.
So, the center of $G$ has finite index.  This implies that the commutator subgroup $[G,G]$ is finite (see this question for instance).  But $G/[G,G]$ is an abelian group generated by finitely many elements of finite order, so it is finite as well.  Thus $G$ is finite, and so the result for finite groups applies.
