TITLE: Number of solutions to $x^n=e$ in group $G$ is divisible by $n$
QUESTION [11 upvotes]: My question is, can we prove that the number of solutions to $x^n=e$ in group $G$ is divisible by $n$, for $n | |G|$ 
Apparently the answer is it is indeed true, infact for the equation $x^n = a$ for any $a \in G $ also, however I have had many ideas each of which having a small flaw with them.
For example I attempted induction on order of $G$ as follows:
Assume true for all groups of order $ \leq k$. Consider $|G| = k+1$.
If $G$ is cyclic, we show fairly easily the result holds. (In fact if I recall correctly there are $n$ solutions for each divisor $n$).
If $G$ is not cyclic, we can look at the subgroups of $G$. It'd be nice to say the number of solutions in each group is divisible by $n$ by the induction hypothesis and get the desired answer by inclusion-exlusion for solutions in all the subgroups, however we could potentially have a subgroup of order  less than $n$ and still contain a solution. 
So if we were able to show it works for all the subgroups of orders $ < n$ collectively we would be done. I have no idea whether this can work though.
I did see this question The number of solutions of $x^n = e$ in a finite group is a multiple of n, whenever n divides the group order. however I couldn't find much help on it. From the hint given I managed to show $no. solutions = \sum_{ d|n} \lambda_{i} \phi(d)$ for some $\lambda_{i}$s, however they could be different so I can't just use the well know summing of euler-phi formula.
Any solution/ideas would be very much appreciated.

REPLY [10 votes]: This is precisely the Theorem of Frobenius.
Without representation theory, there is a proof, by induction; but more - it is double induction (on order of group and also on $n$). I didn't feel the proof elementary, so I am not writing proof here. But you may see following.
This theorem appears in very few standard books of Group Theory (Zassenhaus, M. Hall, Huppert, and no other books I saw giving group-theoretic proof).
However, a group-theoretic proof has been appeared in an interesting article in American Mathematical Monthly: http://www.maa.org/sites/default/files/3004416017467.pdf.bannered.pdf
The interesting things about this article is that it involves some niceapplications of it, and also gives all the sources of earlier known proofs. 
Hope this would help! Let me know!