The p-group fixed point theorem

The goal of this post is to collect a list of applications of the following theorem, which is perhaps the simplest example of a fixed point theorem.

Theorem: Let Image may be NSFW.
Clik here to view. be a finite Image may be NSFW.
Clik here to view.-group acting on a finite set Image may be NSFW.
Clik here to view.. Let Image may be NSFW.
Clik here to view. X^G denote the subset of Image may be NSFW.
Clik here to view. consisting of those elements fixed by Image may be NSFW.
Clik here to view.. Then Image may be NSFW.
Clik here to view. $|X^G| \equiv |X| \bmod p$ ; in particular, if Image may be NSFW.
Clik here to view. $p \nmid |X|$ then Image may be NSFW.
Clik here to view. has a fixed point.

Although this theorem is an elementary exercise, it has a surprising number of fundamental corollaries.

Proof

Image may be NSFW.
Clik here to view. is a disjoint union of orbits for the action of Image may be NSFW.
Clik here to view., all of which have the form Image may be NSFW.
Clik here to view. G/H and hence all of which have cardinality divisible by Image may be NSFW.
Clik here to view. except for the trivial orbits corresponding to fixed points.

Some group-theoretic applications

Theorem: Let Image may be NSFW.
Clik here to view. be a finite Image may be NSFW.
Clik here to view.-group. Then its center Image may be NSFW.
Clik here to view. Z(G) is nontrivial.

Corollary: Every finite Image may be NSFW.
Clik here to view.-group is nilpotent.

Proof. Image may be NSFW.
Clik here to view. acts by conjugation on Image may be NSFW.
Clik here to view. $G \setminus \{ 1 \}$ . If Image may be NSFW.
Clik here to view. |G| = p^n , this set has cardinality Image may be NSFW.
Clik here to view. p^n - 1 , which is not divisible by Image may be NSFW.
Clik here to view.. Hence Image may be NSFW.
Clik here to view. has a fixed point, which is precisely a nontrivial central element. Image may be NSFW.
Clik here to view. $\Box$

Cauchy’s theorem: Let Image may be NSFW.
Clik here to view. be a finite group. Suppose Image may be NSFW.
Clik here to view. is a prime dividing Image may be NSFW.
Clik here to view. |G| . Then Image may be NSFW.
Clik here to view. has an element of order Image may be NSFW.
Clik here to view..

Proof. Image may be NSFW.
Clik here to view. $\mathbb{Z}/p\mathbb{Z}$ acts by cyclic shifts on the set

Image may be NSFW.
Clik here to view. $\{ (g_1, g_2, ... g_p) \in G^p : g_1 g_2 ... g_p = 1 \} \setminus \{ (1, 1, ... 1) \in G^p \}$

since if Image may be NSFW.
Clik here to view. g_1 g_2 ... g_p = 1 then Image may be NSFW.
Clik here to view. $g_p (g_1 g_2 ... g_p) g_p^{-1} = 1$ . This set has cardinality Image may be NSFW.
Clik here to view. $|G|^{p-1} - 1$ , which is not divisible by Image may be NSFW.
Clik here to view., hence Image may be NSFW.
Clik here to view. $\mathbb{Z}/p\mathbb{Z}$ has a fixed point, which is precisely a nontrivial element Image may be NSFW.
Clik here to view. $g \in G$ such that Image may be NSFW.
Clik here to view. g^p = 1 . Image may be NSFW.
Clik here to view. $\Box$

Some number-theoretic applications

Fermat’s little theorem: Let Image may be NSFW.
Clik here to view. be a non-negative integer and Image may be NSFW.
Clik here to view. be a prime. Then Image may be NSFW.
Clik here to view. $a^p \equiv a \bmod p$ .

Proof. Image may be NSFW.
Clik here to view. $\mathbb{Z}/p\mathbb{Z}$ acts by cyclic shifts on the set Image may be NSFW.
Clik here to view. [a]^p , where Image may be NSFW.
Clik here to view. $[a] = \{ 1, 2, ... a \}$ ; equivalently, on the set of strings of length Image may be NSFW.
Clik here to view. on Image may be NSFW.
Clik here to view. letters. (Orbits of this group action are sometimes called necklaces; see, for example, this previous blog post.) This set has cardinality Image may be NSFW.
Clik here to view. a^p and its fixed point set has cardinality Image may be NSFW.
Clik here to view., since a function is fixed if and only if it is constant. Image may be NSFW.
Clik here to view. $\Box$

Fermat’s little theorem for matrices: Let Image may be NSFW.
Clik here to view. be a square matrix of non-negative integers and let Image may be NSFW.
Clik here to view. be a prime. Then Image may be NSFW.
Clik here to view. $\text{tr}(A^p) \equiv \text{tr}(A) \bmod p$ .

This result also appeared in this previous blog post.

Proof. Interpret Image may be NSFW.
Clik here to view. as the adjacency matrix of a graph. Image may be NSFW.
Clik here to view. $\mathbb{Z}/p\mathbb{Z}$ acts by cyclic shifts on the set of closed walks of length Image may be NSFW.
Clik here to view. on this graph. This set has cardinality Image may be NSFW.
Clik here to view. $\text{tr}(A^p)$ and its fixed point set has cardinality Image may be NSFW.
Clik here to view. $\text{tr}(A)$ , since a closed walk is fixed if and only if it consists of Image may be NSFW.
Clik here to view. repetitions of the same loop at a vertex. Image may be NSFW.
Clik here to view. $\Box$

In the same way that Fermat’s little theorem can be used to construct a primality test, the Fermat primality test, Fermat’s little theorem for matrices can also be used to construct a primality test, which doesn’t seem to have a name. For example, the Perrin pseudoprimes are the numbers that pass this test when

Image may be NSFW.
Clik here to view. $\displaystyle A = \left[ \begin{array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 1 & 0 \end{array} \right]$ .

More generally, a stronger version of this test gives rise to the notion of Frobenius pseudoprime.

Wilson’s theorem: Let Image may be NSFW.
Clik here to view. be a prime. Then Image may be NSFW.
Clik here to view. $(p-1)! \equiv -1 \bmod p$ .

Proof. Consider the set of total orderings of the numbers Image may be NSFW.
Clik here to view. $\{ 0, 1, 2, ... p-1 \}$ modulo addition Image may be NSFW.
Clik here to view. $\bmod p$ ; that is, we identify the total ordering

Image may be NSFW.
Clik here to view. $\displaystyle a_0 < a_1 < ... < a_{p-1}$

with the total ordering

Image may be NSFW.
Clik here to view. $\displaystyle a_0 + k < a_1 + k < ... < a_{p-1} + k$

where the addition occurs Image may be NSFW.
Clik here to view. $\bmod p$ .

Image may be NSFW.
Clik here to view. $\mathbb{Z}/p\mathbb{Z}$ acts by cyclic shifts on this set. It has cardinality Image may be NSFW.
Clik here to view. (p-1)! and its fixed point set has cardinality Image may be NSFW.
Clik here to view. p-1 , since the fixed orderings are precisely the ones such that Image may be NSFW.
Clik here to view. $a_{i+1} \equiv a_i + k \bmod p$ for some Image may be NSFW.
Clik here to view. $k \in \{ 1, 2, ... p-1 \}$ . Image may be NSFW.
Clik here to view. $\Box$

Lucas’ theorem: Let Image may be NSFW.
Clik here to view. m, n be non-negative integers and Image may be NSFW.
Clik here to view. be a prime. Suppose the base-Image may be NSFW.
Clik here to view. expansions of Image may be NSFW.
Clik here to view. m, n are

Image may be NSFW.
Clik here to view. $\displaystyle m = \sum_i m_i p^i, n = \sum n_i p^i$

respectively. Then

Image may be NSFW.
Clik here to view. $\displaystyle {m \choose n} \equiv \prod_i {m_i \choose n_i} \bmod p$ .

Proof. It suffices to show that

Image may be NSFW.
Clik here to view. $\displaystyle {m_0 + p m_1 \choose n_0 + p n_1} \equiv {m_0 \choose n_0} {m_1 \choose n_1} \bmod p$

where Image may be NSFW.
Clik here to view. $0 \le m_0, n_0 < p$ , and then the desired result follows by induction. To see this, consider a set Image may be NSFW.
Clik here to view. of size Image may be NSFW.
Clik here to view. m_0 + p m_1 , and partition it into a block of size Image may be NSFW.
Clik here to view. m_0 together with Image may be NSFW.
Clik here to view. blocks of size Image may be NSFW.
Clik here to view. m_1 . Identify all of the Image may be NSFW.
Clik here to view. blocks of size Image may be NSFW.
Clik here to view. m_1 with each other in some way. Then Image may be NSFW.
Clik here to view. $\mathbb{Z}/p\mathbb{Z}$ acts by cyclic shift on these blocks. This action extends to an action on the set of subsets of Image may be NSFW.
Clik here to view. of size Image may be NSFW.
Clik here to view. n_0 + p n_1 . A fixed point of this action consists of a subset of the block of size Image may be NSFW.
Clik here to view. m_0 together with Image may be NSFW.
Clik here to view. copies of a subset of any block of size Image may be NSFW.
Clik here to view. m_1 . Since the union of the Image may be NSFW.
Clik here to view. copies has size divisible by Image may be NSFW.
Clik here to view., and since Image may be NSFW.
Clik here to view. m_0 < p , it follows that there must be Image may be NSFW.
Clik here to view. n_0 elements in the block of size Image may be NSFW.
Clik here to view. m_0 and Image may be NSFW.
Clik here to view. n_1 elements in each of the Image may be NSFW.
Clik here to view. blocks. Thus the set of subsets has cardinality Image may be NSFW.
Clik here to view. ${m_0 + p m_1 \choose n_0 + p n_1}$ , and its fixed point set has cardinality Image may be NSFW.
Clik here to view. ${m_0 \choose n_0} {m_1 \choose n_1}$ . Image may be NSFW.
Clik here to view. $\Box$

(It is also possible to prove this result directly by considering a more complicated Image may be NSFW.
Clik here to view.-group action. The relevant Image may be NSFW.
Clik here to view.-group is the Sylow Image may be NSFW.
Clik here to view.-subgroup of the symmetric group of Image may be NSFW.
Clik here to view.. It is an iterated wreath product of the cyclic groups Image may be NSFW.
Clik here to view. $\mathbb{Z}/p\mathbb{Z}$ , and describing how it acts on Image may be NSFW.
Clik here to view. requires recursively partitioning blocks in a way that is somewhat tedious to describe.)

The following proof is due to Don Zagier.

Fermat’s two-square theorem: Every prime Image may be NSFW.
Clik here to view. $p \equiv 1 \bmod 4$ is the sum of two squares.

Proof. Let Image may be NSFW.
Clik here to view. be the set of all solutions Image may be NSFW.
Clik here to view. (x, y, z) in the positive integers of the equation Image may be NSFW.
Clik here to view. x^2 + 4yz = p . The involution Image may be NSFW.
Clik here to view. $(x, y, z) \mapsto (x, z, y)$ has the property that its fixed points correspond to solutions to the equation Image may be NSFW.
Clik here to view. x^2 + 4y^2 = p . We will show that such fixed points exist by showing that there are an odd number of them; to do so, it suffices to show that Image may be NSFW.
Clik here to view. |S| is odd, and to show that Image may be NSFW.
Clik here to view. |S| is odd it suffices to exhibit any other involution on Image may be NSFW.
Clik here to view. which has an odd number of fixed points. (Notice that this is two applications of the Image may be NSFW.
Clik here to view.-group fixed point theorem.) We claim that

Image may be NSFW.
Clik here to view. $\displaystyle (x, y, z) \mapsto \begin{cases} (x + 2z, z, y - z - x) & \text{ if } x < y - z \\ (2y - x, y, x - y + z) & \text{ if } y - z < x < 2y \\ (x - 2y, x - y + z, y) & \text{ if } x > 2y \end{cases}$

is an involution on Image may be NSFW.
Clik here to view. with exactly one fixed point, which suffices to prove the desired claim. Verifying that it sends Image may be NSFW.
Clik here to view. to Image may be NSFW.
Clik here to view. is straightforward. If Image may be NSFW.
Clik here to view. (x', y', z') is a fixed point, then since Image may be NSFW.
Clik here to view. are all assumed to be positive integers we cannot have Image may be NSFW.
Clik here to view. x' = x' + 2z' , so we are not in the first case. If we are in the second case, we conclude that Image may be NSFW.
Clik here to view. x' = y' , but then Image may be NSFW.
Clik here to view. x' | p , and since Image may be NSFW.
Clik here to view. $x' \le \sqrt{p}$ we conclude that Image may be NSFW.
Clik here to view. x' = y' = 1 , which gives Image may be NSFW.
Clik here to view. $z' = \frac{p-1}{4}$ . If we are in the third case, we conclude that Image may be NSFW.
Clik here to view. y' = z' , then that Image may be NSFW.
Clik here to view. x' = y' , but this contradicts Image may be NSFW.
Clik here to view. x' > 2y' . Hence Image may be NSFW.
Clik here to view. $(1, 1, \frac{p-1}{4})$ is the unique fixed point as desired. Image may be NSFW.
Clik here to view. $\Box$

The following proof is due to V. Lebesgue.

Quadratic reciprocity: Let Image may be NSFW.
Clik here to view. p, q be odd primes. Let Image may be NSFW.
Clik here to view. $\left( \frac{p}{q} \right)$ denote the Legendre symbol. Then

Image may be NSFW.
Clik here to view. $\displaystyle \left( \frac{p}{q} \right) = \left( \frac{q}{p} \right) (-1)^{ \frac{(p-1)(q-1)}{4} }$ .

Proof. Let Image may be NSFW.
Clik here to view. be an odd prime and consider the set

Image may be NSFW.
Clik here to view. $\displaystyle S_n(p, a) = \{ (x_1, ... x_n) \in \mathbb{F}_p^n : x_1^2 + ... + x_n^2 = a \}$ .

For Image may be NSFW.
Clik here to view. an odd prime, we will compute Image may be NSFW.
Clik here to view. $|S_q(p, 1)| \bmod q$ in two different ways. The first way is to observe that Image may be NSFW.
Clik here to view. $\mathbb{Z}/q\mathbb{Z}$ acts by cyclic shifts on Image may be NSFW.
Clik here to view.. The number of fixed points is the number of Image may be NSFW.
Clik here to view. $x \in \mathbb{F}_p$ such that Image may be NSFW.
Clik here to view. qx^2 = 1 , hence

Image may be NSFW.
Clik here to view. $\displaystyle |S_q(p, 1)| \equiv 1 + \left( \frac{q}{p} \right) \bmod q$ .

The second way is to find a recursive formula for Image may be NSFW.
Clik here to view. |S_n(p, 1)| which will end up being expressed in terms of Image may be NSFW.
Clik here to view. $\left( \frac{p}{q} \right)$ . The details are as follows.

First, it is clear that Image may be NSFW.
Clik here to view. $|S_1(p, a)| = 1 + \left( \frac{a}{p} \right)$ . For Image may be NSFW.
Clik here to view. nonzero, we claim that Image may be NSFW.
Clik here to view. |S_2(p, a)| is independent of Image may be NSFW.
Clik here to view.. First, observe that by the pigeonhole principle (there are Image may be NSFW.
Clik here to view. $\frac{p+1}{2}$ possible values of Image may be NSFW.
Clik here to view. x_1^2 and Image may be NSFW.
Clik here to view. $\frac{p+1}{2}$ possible values of Image may be NSFW.
Clik here to view. a - x_2^2 , hence at least one common value) we always have Image may be NSFW.
Clik here to view. |S_2(p, a)| > 0 . Second, observe that the map Image may be NSFW.
Clik here to view. $(x_1, x_2) \mapsto x_1^2 + x_2^2$ is the norm map from Image may be NSFW.
Clik here to view. $\mathbb{F}_p[i]/(i^2 + 1)$ to Image may be NSFW.
Clik here to view. $\mathbb{F}_p$ ; in particular, it restricts to a homomorphism from the unit group of the former ring to the unit group of the latter ring, and we know that this homomorphism is surjective, so each fiber has the same size. The unit group of Image may be NSFW.
Clik here to view. $\mathbb{F}_p[i]/(i^2 + 1)$ has size Image may be NSFW.
Clik here to view. p^2 - 1 if Image may be NSFW.
Clik here to view. $p \equiv 3 \bmod 4$ (since then we are looking at Image may be NSFW.
Clik here to view. $\mathbb{F}_{p^2}$ ) and size Image may be NSFW.
Clik here to view. (p - 1)^2 if Image may be NSFW.
Clik here to view. $p \equiv 1 \bmod 4$ (since then we are looking at Image may be NSFW.
Clik here to view. $\mathbb{F}_p \times \mathbb{F}_p$ ). Hence

Image may be NSFW.
Clik here to view. $\displaystyle |S_2(p, a)| = p - (-1)^{\frac{p-1}{2}}$

for any nonzero Image may be NSFW.
Clik here to view.. When Image may be NSFW.
Clik here to view. a = 0 , the equation Image may be NSFW.
Clik here to view. x_1^2 + x_2^2 = 0 always has the solution Image may be NSFW.
Clik here to view. (0, 0) . There are no nonzero solutions if Image may be NSFW.
Clik here to view. $p \equiv 3 \bmod 4$ (since Image may be NSFW.
Clik here to view. does not have a square root) and Image may be NSFW.
Clik here to view. 2p - 2 nonzero solutions if Image may be NSFW.
Clik here to view. $p \equiv 1 \bmod 4$ (since for each of the Image may be NSFW.
Clik here to view. p-1 possible values of Image may be NSFW.
Clik here to view. x_1 there are Image may be NSFW.
Clik here to view. possible values of Image may be NSFW.
Clik here to view. x_2 ). Hence

Image may be NSFW.
Clik here to view. $\displaystyle |S_2(p, 0)| = p + (p-1)(-1)^{ \frac{p-1}{2} }$ .

We will now compute Image may be NSFW.
Clik here to view. |S_n(p, a)| inductively by rewriting Image may be NSFW.
Clik here to view. x_1^2 + ... + x_n^2 = a as

Image may be NSFW.
Clik here to view. $\displaystyle x_1^2 + ... + x_{n-2}^2 = a - x_{n-1}^2 - x_n^2$ .

From this it follows that

Image may be NSFW.
Clik here to view. $\displaystyle |S_n(p, a)| = \sum_{x_{n-1}, x_n = 0}^{p-1} |S_{n-2}(p, a - x_{n-1}^2 - x_n^2)|$ .

We know that Image may be NSFW.
Clik here to view. $a - x_{n-1}^2 - x_n^2$ takes the value Image may be NSFW.
Clik here to view. for Image may be NSFW.
Clik here to view. $p + (p-1)(-1)^{ \frac{p-1}{2} }$ possible values of Image may be NSFW.
Clik here to view. $(x_{n-1}, x_n)$ and takes any particular other value for Image may be NSFW.
Clik here to view. $p - (-1)^{ \frac{p-1}{2} }$ possible values of Image may be NSFW.
Clik here to view. $(x_{n-1}, x_n)$ . Hence we can write

Image may be NSFW.
Clik here to view. $\displaystyle |S_n(p, a)| = \left( p - (-1)^{ \frac{p-1}{2} } \right) \sum_{b=0}^{p-1} |S_{n-2}(p, b)| + p (-1)^{ \frac{p-1}{2} } |S_{n-2}(p, a)|.$

But it’s clear that

Image may be NSFW.
Clik here to view. $\displaystyle \sum_{b=0}^{p-1} |S_{n-2}(p, b)| = p^{n-2}$

since this is just the number of possible tuples Image may be NSFW.
Clik here to view. $(x_1, ... x_{n-2})$ . Hence

Image may be NSFW.
Clik here to view. $\displaystyle |S_n(p, a)| = p^{n-1} - p^{n-2} (-1)^{ \frac{p-1}{2} } + p (-1)^{ \frac{p-1}{2} } |S_{n-2}(p, a)|$ .

This gives a recursion which we can unwind into an explicit formula for Image may be NSFW.
Clik here to view. |S_n(p, a)| as follows. Substituting the recursion into itself and canceling terms gives

Image may be NSFW.
Clik here to view. $\displaystyle |S_n(p, a)| = p^{n-1} - p^{n-3} (-1)^{ 2 \frac{p-1}{2} } + p^2 (-1)^{ 2 \frac{p-1}{2} } |S_{n-4}(p, a)|$ .

This pattern continues, and by induction we can show that

Image may be NSFW.
Clik here to view. $\displaystyle |S_n(p, a)| = p^{n-1} - p^{n-1-k} (-1)^{ k \frac{p-1}{2} } + p^k (-1)^{ k \frac{p-1}{2} } |S_{n-2k}(p, a)|$ .

Hence if Image may be NSFW.
Clik here to view. n = 2k+1 is odd, we conclude that

Image may be NSFW.
Clik here to view. $\displaystyle |S_{2k+1}(p, 1)| = p^{2k} + p^k (-1)^{ k \frac{p-1}{2} }$ .

Now let Image may be NSFW.
Clik here to view. 2k+1 = q be an odd prime. Then Image may be NSFW.
Clik here to view. $p^{2k} = p^{q-1} \equiv 1 \bmod q$ by Fermat’s little theorem and Image may be NSFW.
Clik here to view. $p^k = p^{\frac{q-1}{2}} \equiv \left( \frac{p}{q} \right) \bmod q$ by Euler’s criterion, so we we find that

Image may be NSFW.
Clik here to view. $\displaystyle |S_q(p, 1)| \equiv 1 + \left( \frac{p}{q} \right) (-1)^{ \frac{(p-1)(q-1)}{4} } \bmod q$ .

On the other hand, we know that Image may be NSFW.
Clik here to view. $|S_q(p, 1)| \equiv 1 + \left( \frac{q}{p} \right) \bmod q$ . We conclude that

Image may be NSFW.
Clik here to view. $\displaystyle \left( \frac{q}{p} \right) \equiv \left( \frac{p}{q} \right) (-1)^{ \frac{(p-1)(q-1)}{4} } \bmod q$

and since the LHS and RHS are both equal to Image may be NSFW.
Clik here to view. $\pm 1$ this equality holds on the nose. Image may be NSFW.
Clik here to view. $\Box$

This proof is in some sense a proof via Gauss sums in disguise; the numbers Image may be NSFW.
Clik here to view. |S_n(p, a)| are closely related to Jacobi sums, which can be written in terms of Gauss sums. More generally, various kinds of exponential sums can be related to counting points on varieties over finite fields. The Weil conjectures give bounds on the latter which translate into bounds on the former, and this has various applications, for example the original proof of Zhang’s theorem on bounded gaps between primes (although my understanding is that this step is unnecessary just to establish the theorem).

Similar methods can be used to prove the second supplementary law, although strictly speaking we will use more than the Image may be NSFW.
Clik here to view.-group fixed point theorem.

Quadratic reciprocity, second supplement: Image may be NSFW.
Clik here to view. $\displaystyle \left( \frac{2}{p} \right) = (-1)^{ \frac{p^2-1}{8} }$ . Equivalently, Image may be NSFW.
Clik here to view. $\displaystyle \left( \frac{2}{p} \right) = 1$ iff Image may be NSFW.
Clik here to view. $p \equiv \pm 1 \bmod 8$ .

Proof. We will compute Image may be NSFW.
Clik here to view. $|S_2(p, 1)| \bmod 8$ in two different ways. The first is using the formula Image may be NSFW.
Clik here to view. $|S_2(p, 1)| = p - (-1)^{ \frac{p-1}{2} }$ we found earlier, which gives

Image may be NSFW.
Clik here to view. $\displaystyle |S_2(p, 1)| \equiv \begin{cases} 0 \bmod 8 \text{ if } p \equiv 1 \bmod 8 \\ 4 \bmod 8 \text{ if } p \equiv 3 \bmod 8 \\ 4 \bmod 8 \text{ if } p \equiv 5 \bmod 8 \\ 0 \bmod 8 \text{ if } p \equiv 7 \bmod 8 \\ \end{cases}$ .

The second is to observe that the dihedral group Image may be NSFW.
Clik here to view. D_4 of order Image may be NSFW.
Clik here to view. acts on Image may be NSFW.
Clik here to view. S_2(p, 1) by rotation and reflection (thinking of the points as being on the “circle” defined by Image may be NSFW.
Clik here to view. x_1^2 + x_2^2 = 1 ). Most orbits of this action have size Image may be NSFW.
Clik here to view. and hence are invisible Image may be NSFW.
Clik here to view. $\bmod 8$ . The ones that are not invisible Image may be NSFW.
Clik here to view. $\bmod 8$ correspond to points with some nontrivial stabilizer. Any stabilizer contains an element of order Image may be NSFW.
Clik here to view., and there are five such elements in Image may be NSFW.
Clik here to view. D_4 , so the remaining possible orbits are as follows:

Points stable under Image may be NSFW.
Clik here to view. $(x, y) \mapsto (y, x)$ . These are precisely the points Image may be NSFW.
Clik here to view. such that Image may be NSFW.
Clik here to view., hence there are Image may be NSFW.
Clik here to view. $1 + \left( \frac{2}{p} \right)$ of them.
Points stable under Image may be NSFW.
Clik here to view. $(x, y) \mapsto (x, -y)$ . These are precisely the points Image may be NSFW.
Clik here to view. such that Image may be NSFW.
Clik here to view., hence there are Image may be NSFW.
Clik here to view. of them.
Points stable under Image may be NSFW.
Clik here to view. $(x, y) \mapsto (-x, y)$ . These are precisely the points Image may be NSFW.
Clik here to view. such that Image may be NSFW.
Clik here to view., hence there are Image may be NSFW.
Clik here to view. of them.
Points stable under Image may be NSFW.
Clik here to view. $(x, y) \mapsto (-x, -y)$ . There are no such points.
Points stable under Image may be NSFW.
Clik here to view. $(x, y) \mapsto (-y, -x)$ . These are precisely the points Image may be NSFW.
Clik here to view. such that Image may be NSFW.
Clik here to view., hence there are Image may be NSFW.
Clik here to view. $1 + \left( \frac{2}{p} \right)$ of them.

In total, we conclude that

Image may be NSFW.
Clik here to view. $\displaystyle 6 + 2 \left( \frac{2}{p} \right) \equiv \begin{cases} 0 \bmod 8 \text{ if } p \equiv 1 \bmod 8 \\ 4 \bmod 8 \text{ if } p \equiv 3 \bmod 8 \\ 4 \bmod 8 \text{ if } p \equiv 5 \bmod 8 \\ 0 \bmod 8 \text{ if } p \equiv 7 \bmod 8 \\ \end{cases}$

and the conclusion follows. Image may be NSFW.
Clik here to view. $\Box$

A general lesson of the above proofs is that it is a good idea to replace integers with sets because sets can be equipped with more structure, such as group actions, than integers. This is a simple form of categorification.

The p-group fixed point theorem

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