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 $(p-1)! \equiv -1 \mod p$. 



Applications of Frobenius theorem and conjecture
Danny Gorenstein, in his book *Finite Simple Groups, An Introduction to their Classification* discusses a very significant application of Frobenius' theorem (see p. 95). He is interested in the following central problem in the Classification of Finite Simple Groups:



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> Let $G$ be a simple group in which the structure of the centralizer of an involution is given. Determine the order of $G$.
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This problem naturally splits into different cases, the most difficult of which concerns the case where $G$ has a single conjugacy class of involutions. One proceeds by studying the $2$-local structure of $G$, then the $p$-local structure, for those primes $p$ dividing $|X|$ where $X$ is the given involution centralizer.


Gorenstein remarks that



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> With this information, one can now obtain a *congruence* for the order of $G$ with the aid of Sylow's theorem and a result of Frobenius.
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The `result of Frobenius' he refers to is the one in this question. I'm unqualified to write more so I recommend Gorenstein's book for more details (on this and anything else to do with CFSG).


One final remark: Gorenstein notes in a footnote that Hall's book *The Theory of Groups* contains a strengthening of Frobenius' result (see Theorem 9.1.1, I'm not sure if this is the same generalization mentioned by Anton above):



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> If $C$ is a conjugacy class of elements of a finite group $G$, then for any positive integer $n$, the number of solutions in $G$ of the equation $x^n=c, c\in C$, is a multiple of ${\rm gcd}(|C|n, |G|)$.
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