TITLE: Applications of Frobenius theorem and conjecture
QUESTION [35 upvotes]: A theorem of Frobenius states that if $n$ divides the order of a finite group $G$, then the number of solutions to $x^n = 1$ in $G$ is a multiple of $n$. Frobenius conjectured that if the number of solutions is exactly $n$, then the set of solutions form a characteristic subgroup of $G$. The conjecture was eventually proved in the 90's, and the full proof uses the classification of finite simple groups.
The theorem feels a bit isolated for me.. I'm not sure how the conjecture fits into a wider context, either. What is their importance, if any? Are there any good examples of applications of the theorem or the conjecture? If I'm interested in finite groups, why should I care about the theorem or the conjecture, other than that they are kind of neat?
One example I know is that if $G$ has every Sylow subgroup cyclic, then with Frobenius theorem we can show that the Sylow subgroup corresponding to the largest prime divisor of $G$ is normal. Also (this one is too easy, but I like it) for any prime $p$, the number of elements satisfying $x^p = 1$ in the symmetric group $S_p$ is $(p-1)! + 1$, so Frobenius theorem implies $(p-1)! \equiv -1 \mod p$.

REPLY [12 votes]: An application of this theorem to $p$-adic analysis is the $p$-integrality of the coefficients of the Artin-Hasse exponential
$$
{\rm AH}_p(X) = e^{X + X^p/p + X^{p^2}/p^2 + \cdots}.
$$
That means when this series is expanded as $\sum_{k \geq 0} a_kX^k$, the coefficients $a_k$ don't have their denominator divisible by $p$. This is obvious for $k = 0$ since $a_0 = 1$.  We have $a_k = 1/k!$ for $1 \leq k \leq p-1$, so $a_1,\dots,a_{p-1}$ are all $p$-integral. And $a_p = ((p-1)!+1)/p!,$ which is $p$-integral by Wilson's theorem. To handle the general case we'll use the Frobenius theorem.
For $k \geq 1$, a non-obvious combinatorial/probabilistic formula for the coefficients is 
$$
a_k = \frac{\#\{g \in S_k : g \text{ has } p\text{-power order}\}}{k!}.
$$
Using the Frobenius theorem with $n$ being the largest power of $p$ dividing $|S_k|$, the numerator of $a_k$ is divisible by the largest power of $p$ in the denominator. Hence each $a_k$ is $p$-integral.
There are simpler proofs of the $p$-integrality of the coefficients of this series. See the Wikipedia page on the Artin-Hasse exponential.