Let $G$ be a finite group and $p$ a prime dividing $|G|$. There is a nice proof of Cauchy's Theorem in Aluffi's "Algebra Chapter 0" that makes use of an action (by cyclic permutation) of $\mathbb{Z}/p\mathbb{Z}$ on the set of tuples $(x_1,\ldots,x_p)$ with $x_i\in G$ such that $x_1\cdots x_p=1$. Essentially, the argument shows that the set of fixed points of this action, namely, the collection of tuples of the form $(x,\ldots,x)$ with $x^p=1$, must have size divisible by $p$, and since $(1,\ldots,1)$ is in there, we get an element of order $p$.
I noticed that this argument can be extended to prove the following: If $G$ is a finite group, $p$ divides $|G|$, $\emptyset\neq H\subseteq Z(G)$, and $N$ denotes the number of $x\in G$ with $x^p\in H$, then $p$ divides $N$. For the proof, consider the set of tuples $S=\{(x_1,\ldots,x_p): x_i\in G, x_1\cdots x_p\in H\}$. We can count the number of elements of $S$ as follows: For any $h\in H$, we may choose $x_1,\ldots,x_{p-1}$ arbitrarily before $x_p=(x_1\cdots x_{p-1})^{-1}h$ is determined. This shows (I think) that $|S|=|G|^{p-1}|H|$, a multiple of $p$. Now, if $(x_1,\ldots,x_p)\in S$, then $x_1\cdots x_p=h$ for some $h\in H$. Since $h\in Z(G)$, we have $x_2\cdots x_p=x_1^{-1}h=hx_1^{-1}\implies x_2\cdots x_px_1=h\implies (x_2,\ldots,x_p,x_1)\in S$. So, we get an action of $\mathbb{Z}/p\mathbb{Z}$ on $S$ by cyclic permutation, as in the proof of Cauchy's Theorem. Again, the fixed points are the "constant" tuples in $S$, and the formula $|S|=|\{$fixed points$\}|+\sum\{$sizes of nontrivial orbits$\}$ shows that $p$ divides $N$.
Of course, we can now quickly deduce Cauchy's Theorem by setting $H=\{1\}$ and noting that since $1^p=1$, $N\geq p$.
My questions are: Is the above result well-known? Is it interesting? Am I overcomplicating something simple?
Best Answer
I don't know if I would call your result "well-known", but it is a special case of some other results which are at least known.
Let $G$ be a finite group. Frobenius' theorem (1895) states the following.
More generally, Frobenius (1903) proved:
(There is a proof in "Theory of Finite Groups" by Marshall Hall Jr., Theorem 9.1.1.)
As a corollary:
So if $p \mid |G|$ is a prime and $c \in Z(G)$, the number of solutions to $x^p = c$ in $G$ is a multiple of $p$. Thus giving your observation, although the proofs of the above results are different. The proof of Cauchy's theorem that you mention is due to McKay (1959), and basically proves Frobenius' theorem for $n = p$ prime.
These theorems by Frobenius have been generalized by many authors, and there are many different proofs. For example, Philip Hall proved many results on the number of solutions to more general equations in finite groups.
EDIT: We could generalize the argument of McKay that appears in your question as follows. Let $p$ be a prime dividing $|G|$.
Suppose that $X \subseteq G$ is a subset of $G$ which is closed under conjugation. Consider $$S = \{ (x_1, \ldots, x_p) : x_1 \cdots x_p \in X \}.$$ Then $|S| = |X| |G|^{p-1}$, and $\mathbb{Z}/p\mathbb{Z}$ acts on $S$ by cyclic permutation. Therefore the number of elements with $x^p \in X$ is a multiple of $p$. From this we get a special case of Frobenius' theorem.
In particular we have a proof of Frobenius' theorem in the case where $n$ is squarefree. The above proof is basically the one given by Sachs (1960).
References:
[1] F. G. Frobenius, Verallgemeinerung des Sylow'schen Satzes, Sitzungsberichte der Königl. Preuß. Akad. der Wissenschaften (Berlin) (1895), 981-993.
[2] F. G. Frobenius, Über einen Fundamentalsatz der Gruppentheorie, Sitzungsberichte der Königl. Preuß. Akad. der Wissenschaften (Berlin) (1903), 987-991.
[3] P. Hall, On a Theorem of Frobenius, Proc. London Math. Soc., 40, 468-501, (1935).
[4] J. H. McKay, Another proof of Cauchy's group theorem. Amer. Math. Monthly 66 (1959), 119.
[5] H. Sachs, Einfacher Beweis des Frobeniusschen Fundamentalsatzes der Gruppentheorie für den Fall eines quadratfreien Exponenten. Acta Sci. Math. (Szeged) 21 (1960), 309-310.