Meditation on the Sylow theorems I

As an undergraduate the proofs I saw of the Sylow theorems seemed very complicated and I was totally unable to remember them. The goal of this post is to explain proofs of the Sylow theorems which I am actually able to remember, several of which use our old friend

The Image may be NSFW.
Clik here to view.-group fixed point theorem (PGFPT): If Image may be NSFW.
Clik here to view. is a finite Image may be NSFW.
Clik here to view.-group and Image may be NSFW.
Clik here to view. is a finite set on which Image may be NSFW.
Clik here to view. acts, then the subset Image may be NSFW.
Clik here to view. X^P of fixed points satisfies Image may be NSFW.
Clik here to view. $|X^P| \equiv |X| \bmod p$ . In particular, if Image may be NSFW.
Clik here to view. $|X| \not \equiv 0 \bmod p$ then this action has at least one fixed point.

There will be some occasional historical notes taken from Waterhouse’s The Early Proofs of Sylow’s Theorem.

A quick note on the definition of a Sylow subgroup

A Sylow Image may be NSFW.
Clik here to view.-subgroup of a finite group Image may be NSFW.
Clik here to view. can be equivalently defined as either a Image may be NSFW.
Clik here to view.-subgroup of index coprime to Image may be NSFW.
Clik here to view. |G| (that is, if Image may be NSFW.
Clik here to view. |G| = p^n k where Image may be NSFW.
Clik here to view. $\gcd(p, k) = 1$ , then Image may be NSFW.
Clik here to view. |P| = p^n ) or a Image may be NSFW.
Clik here to view.-subgroup which is maximal with respect to inclusion. The equivalence of these two definitions requires the Sylow theorems, but we’ll want to discuss such subgroups before proving them, and for the purposes of this post it will be most convenient to adopt the first definition.

So for the rest of this post, a Sylow Image may be NSFW.
Clik here to view.-subgroup is a Image may be NSFW.
Clik here to view.-subgroup of index coprime to Image may be NSFW.
Clik here to view., and we’ll refer to Image may be NSFW.
Clik here to view.-subgroups maximal with respect to inclusion as maximal Image may be NSFW.
Clik here to view.-subgroups before we prove that the two are equivalent.

Warmup: Cauchy’s theorem

We’ll start off by giving some proofs of Cauchy’s theorem. One of the proofs of Sylow I later needs it as a lemma, and some of the proofs we give here will generalize to proofs of Sylow I. First we quote the proof using the PGFPT in full from the above blog post, as Proof 1.

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 1 (PGFPT). 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. $\Box$

This proof is almost disgustingly short. By contrast, according to Waterhouse, Cauchy’s original proof runs nine pages (!) and works as follows.

First Cauchy explicitly constructs Sylow Image may be NSFW.
Clik here to view.-subgroups of the symmetric group Image may be NSFW.
Clik here to view. S_n . This can be done by recursively partitioning Image may be NSFW.
Clik here to view. $\{ 1, 2, \dots n \}$ into Image may be NSFW.
Clik here to view. $\lfloor \frac{n}{p} \rfloor$ blocks of size Image may be NSFW.
Clik here to view., considering the subgroup generated by Image may be NSFW.
Clik here to view.-cycles cyclically permuting each block, then partitioning the set of blocks itself into Image may be NSFW.
Clik here to view. $\lfloor \frac{n}{p^2} \rfloor$ blocks-of-blocks of size Image may be NSFW.
Clik here to view., cyclically permuting these, and so forth. This subgroup has size

Image may be NSFW.
Clik here to view. $\displaystyle p^{ \sum_{k \ge 1} \left\lfloor \frac{n}{p^k} \right\rfloor }$

and the exponent is Legendre’s formula for the largest power Image may be NSFW.
Clik here to view. $\nu_p(n!)$ of Image may be NSFW.
Clik here to view. dividing Image may be NSFW.
Clik here to view. |S_n| . It follows that this is a Sylow Image may be NSFW.
Clik here to view.-subgroup.

Second, Cauchy proves a special case of the following lemma, namely the special case that Image may be NSFW.
Clik here to view. H = S_n . This lemma doesn’t appear to have a standard name, so we’ll name it

Cauchy’s Image may be NSFW.
Clik here to view.-group lemma: Let Image may be NSFW.
Clik here to view. be a subgroup of a finite group Image may be NSFW.
Clik here to view. such that Image may be NSFW.
Clik here to view. has a Sylow Image may be NSFW.
Clik here to view.-subgroup Image may be NSFW.
Clik here to view.. If Image may be NSFW.
Clik here to view. $p \mid |G|$ , then Image may be NSFW.
Clik here to view. has an element of order Image may be NSFW.
Clik here to view..

Proof. We’ll prove the contrapositive: if Image may be NSFW.
Clik here to view. doesn’t contain an element of order Image may be NSFW.
Clik here to view., then Image may be NSFW.
Clik here to view. does not divide Image may be NSFW.
Clik here to view. |G| .

Consider the left action of Image may be NSFW.
Clik here to view. on the set of cosets Image may be NSFW.
Clik here to view. H/P . By grouping the cosets into Image may be NSFW.
Clik here to view.-orbits and using the orbit-stabilizer theorem we have

Image may be NSFW.
Clik here to view. $\displaystyle |H/P| = \sum_{[h] \in G \backslash H/P} \frac{|G|}{|\text{Stab}_G(hP)|}$

where Image may be NSFW.
Clik here to view. $G \backslash H/P$ is the collection of double cosets and the stabilizer of a coset is

Image may be NSFW.
Clik here to view. $\displaystyle \text{Stab}_G(hP) = \{ g \in G : ghP = hP \} = \{ g \in G : h^{-1} gh \in P \} = h^{-1} G h \cap P$

and in particular it has order dividing Image may be NSFW.
Clik here to view. |P| , hence a power of Image may be NSFW.
Clik here to view..

If Image may be NSFW.
Clik here to view. has no element of order Image may be NSFW.
Clik here to view., then since any nontrivial subgroup of Image may be NSFW.
Clik here to view. contains an element of order Image may be NSFW.
Clik here to view., these stabilizers are all trivial, so we get that

Image may be NSFW.
Clik here to view. $\displaystyle |H/P| = |G| |G \backslash H/P|$

and in particular that Image may be NSFW.
Clik here to view. |G| divides Image may be NSFW.
Clik here to view. |H/P| . Since Image may be NSFW.
Clik here to view. is a Sylow Image may be NSFW.
Clik here to view.-subgroup, Image may be NSFW.
Clik here to view. |H/P| isn’t divisible by Image may be NSFW.
Clik here to view., and hence neither is Image may be NSFW.
Clik here to view. |G| . Image may be NSFW.
Clik here to view. $\Box$

Call a collection of finite groups cofinal if every finite group embeds into at least one of them.

Corollary: To prove Cauchy’s theorem for all finite groups, it suffices to find a cofinal collection of finite groups which have Sylow Image may be NSFW.
Clik here to view.-subgroups.

We can now complete Cauchy’s proof of Cauchy’s theorem.

Proof 2 (symmetric groups). The symmetric groups Image may be NSFW.
Clik here to view. $\{ S_n \}$ are cofinal (this is exactly Cayley’s theorem) and have Sylow Image may be NSFW.
Clik here to view.-subgroups. Image may be NSFW.
Clik here to view. $\Box$

But we don’t have to use the symmetric groups!

Proof 3 (general linear groups). Since the symmetric groups Image may be NSFW.
Clik here to view. S_n embed into the general linear groups Image may be NSFW.
Clik here to view. $GL_n(\mathbb{F}_p)$ , the latter are also cofinal, and we can also exhibit Sylow Image may be NSFW.
Clik here to view.-subgroups for them: the unipotent subgroup Image may be NSFW.
Clik here to view. $U_n(\mathbb{F}_p)$ of upper triangular matrices with Image may be NSFW.
Clik here to view.s on the diagonal is Sylow. This follows from the standard calculation of the order

Image may be NSFW.
Clik here to view. $\displaystyle |GL_n(\mathbb{F}_p)| = p^{ {n \choose 2} } (p-1)^n [n]_p!$

where Image may be NSFW.
Clik here to view. [n]_p! is the Image may be NSFW.
Clik here to view.-factorial. Image may be NSFW.
Clik here to view. $\Box$

Here’s a fourth proof that doesn’t use Cauchy’s lemma and doesn’t require the clever construction of the first proof; Waterhouse says it’s used by Rotman.

Proof 4 (class equation). We induct on the order of Image may be NSFW.
Clik here to view.. First we observe that Cauchy’s theorem is straightforwardly true for finite abelian groups by using the Chinese remainder theorem to write a finite abelian group as a direct sum of finite abelian Image may be NSFW.
Clik here to view.-groups. (And then, to spell it out very explicitly, Lagrange’s theorem implies that any nontrivial element of a finite Image may be NSFW.
Clik here to view.-group has Image may be NSFW.
Clik here to view.-power order, so some power of that element has order Image may be NSFW.
Clik here to view..)

Next, given a finite group Image may be NSFW.
Clik here to view. such that Image may be NSFW.
Clik here to view. $p \mid |G|$ we consider the class equation

Image may be NSFW.
Clik here to view. $\displaystyle |G| = |Z(G)| + \sum_i \frac{|G|}{|C_G(g_i)|}$

where the sum runs over non-central conjugacy classes. If Image may be NSFW.
Clik here to view. $p \mid |Z(G)|$ then we’re done by reduction to the abelian case. Otherwise, Image may be NSFW.
Clik here to view. doesn’t divide Image may be NSFW.
Clik here to view. |Z(G)| but does divide Image may be NSFW.
Clik here to view. |G| , so by taking both sides Image may be NSFW.
Clik here to view. $\bmod p$ we conclude that there is some Image may be NSFW.
Clik here to view. g_i such that the centralizer Image may be NSFW.
Clik here to view. C_G(g_i) has index in Image may be NSFW.
Clik here to view. coprime to Image may be NSFW.
Clik here to view., and hence Image may be NSFW.
Clik here to view. $p \mid |C_G(g_i)|$ . Since Image may be NSFW.
Clik here to view. g_i is a non-central conjugacy class Image may be NSFW.
Clik here to view. has order strictly less than Image may be NSFW.
Clik here to view. |G| so has an element of order Image may be NSFW.
Clik here to view. by the inductive hypothesis. Image may be NSFW.
Clik here to view. $\Box$

Sylow I

With very little additional effort the above proof of Cauchy’s Image may be NSFW.
Clik here to view.-group lemma actually proves the following stronger result; Waterhouse attributes this argument, again in the special case that Image may be NSFW.
Clik here to view. H = S_n , to Frobenius. It also doesn’t appear to have a standard name, so we’ll call it

Frobenius’ Image may be NSFW.
Clik here to view.-group lemma: Let Image may be NSFW.
Clik here to view. be a subgroup of a finite group Image may be NSFW.
Clik here to view. such that Image may be NSFW.
Clik here to view. has a Sylow Image may be NSFW.
Clik here to view.-subgroup Image may be NSFW.
Clik here to view.. Then Image may be NSFW.
Clik here to view. also has a Sylow Image may be NSFW.
Clik here to view.-subgroup, given by its intersection with some conjugate of Image may be NSFW.
Clik here to view..

Proof. Consider as before the sum

Image may be NSFW.
Clik here to view. $\displaystyle |H/P| = \sum_{[h] \in G \backslash H/P} \frac{|G|}{|h^{-1} G h \cap P|}$

obtained by considering the orbits of the action of Image may be NSFW.
Clik here to view. on Image may be NSFW.
Clik here to view. H/P . Since Image may be NSFW.
Clik here to view. doesn’t divide Image may be NSFW.
Clik here to view. |H/P| , at least one term on the RHS is also not divisible by Image may be NSFW.
Clik here to view., so there exists some Image may be NSFW.
Clik here to view. such that Image may be NSFW.
Clik here to view. $h^{-1} G h \cap P$ has index coprime to Image may be NSFW.
Clik here to view. and hence is a Sylow Image may be NSFW.
Clik here to view.-subgroup of Image may be NSFW.
Clik here to view. $h^{-1} G h$ ; conjugating, we get that the intersection of Image may be NSFW.
Clik here to view. $h P h^{-1}$ with Image may be NSFW.
Clik here to view. is a Sylow Image may be NSFW.
Clik here to view.-subgroup of Image may be NSFW.
Clik here to view. as desired. Image may be NSFW.
Clik here to view. $\Box$

Corollary: To prove that Sylow subgroups exist for all finite groups, it suffices to prove that they exist for a cofinal collection of finite groups.

Now we can give two different proofs of Sylow I, as follows.

Sylow I: Every finite group Image may be NSFW.
Clik here to view. has a Sylow Image may be NSFW.
Clik here to view.-subgroup.

(Note that with our definitions if Image may be NSFW.
Clik here to view. $\gcd(p, |G|) = 1$ then this subgroup is just the trivial group.)

Proof 1 (symmetric groups). It suffices to prove the existence of Sylow subgroups for the symmetric groups Image may be NSFW.
Clik here to view. S_n , which was done above. Image may be NSFW.
Clik here to view. $\Box$

Proof 2 (general linear groups). It suffices to prove the existence of Sylow Image may be NSFW.
Clik here to view.-subgroups for Image may be NSFW.
Clik here to view. $GL_n(\mathbb{F}_p)$ , which was done above. Running the argument gives that every finite Image may be NSFW.
Clik here to view.-group is a subgroup of Image may be NSFW.
Clik here to view. $U_n(\mathbb{F}_p)$ for some Image may be NSFW.
Clik here to view., which gives an independent proof that finite Image may be NSFW.
Clik here to view.-groups are nilpotent. Image may be NSFW.
Clik here to view. $\Box$

Proof 2 is the first proof of Sylow I given by Serre in his Finite Groups: An Introduction.

I heard a rumor years ago that these proofs existed but didn’t see them; glad that’s finally settled.

The next proof of Sylow I we’ll present is the one that I saw as an undergraduate, in Artin’s Algebra if memory serves. According to Wikipedia it’s due to Wielandt. Serre attributes it to “Miller-Wielandt” and it’s the second proof he gives.

Proof 3 (action on subsets). Write Image may be NSFW.
Clik here to view. |G| = p^k m where Image may be NSFW.
Clik here to view. $\gcd(p, m) = 1$ and consider the action of Image may be NSFW.
Clik here to view. by left multiplication on the set Image may be NSFW.
Clik here to view. ${G \choose p^k}$ of Image may be NSFW.
Clik here to view. p^k -element subsets of Image may be NSFW.
Clik here to view.. We’ll show that there exists a stabilizer subgroup of size Image may be NSFW.
Clik here to view. p^k .

(I found this construction very bizarre and unmotivated as an undergraduate. I like it better now, I guess, but I’m still a little mystified.)

Key observation: An element Image may be NSFW.
Clik here to view. $g \in G$ stabilizes a subset Image may be NSFW.
Clik here to view. $S \subseteq G$ iff Image may be NSFW.
Clik here to view. consists of cosets of the cyclic subgroup Image may be NSFW.
Clik here to view. $\langle g \rangle$ generated by Image may be NSFW.
Clik here to view.; in particular a necessary condition is that its order must divide the size of Image may be NSFW.
Clik here to view..

This means that the action of Image may be NSFW.
Clik here to view. on Image may be NSFW.
Clik here to view. ${G \choose p^k}$ has the property that all stabilizers must be Image may be NSFW.
Clik here to view.-groups (just like the action of Image may be NSFW.
Clik here to view. on Image may be NSFW.
Clik here to view. H/P we used above). As before, decomposing this action into orbits gives

Image may be NSFW.
Clik here to view. $\displaystyle {|G| \choose p^k} = \sum_{[S]} \frac{|G|}{|\text{Stab}(S)|}$

where the sum runs over a set of orbit representatives.

Lemma: Image may be NSFW.
Clik here to view. $\displaystyle {p^k m \choose p^k} \ \equiv m \bmod p$ .

Proof. This is a special case of Lucas’ theorem , which we proved using the PGFPT previously. Image may be NSFW.
Clik here to view. $\Box$

It follows that Image may be NSFW.
Clik here to view. ${|G| \choose p^k} \equiv m \bmod p$ is not divisible by Image may be NSFW.
Clik here to view., so there exists some subset Image may be NSFW.
Clik here to view. such that Image may be NSFW.
Clik here to view. $\frac{|G|}{|\text{Stab}(S)|}$ is also not divisible by Image may be NSFW.
Clik here to view.. But since Image may be NSFW.
Clik here to view. $|\text{Stab}(S)|$ is a power of Image may be NSFW.
Clik here to view. this is only possible if Image may be NSFW.
Clik here to view. $|\text{Stab}(S)| = p^k$ . This stabilizer is a Sylow Image may be NSFW.
Clik here to view.-subgroup as desired. Image may be NSFW.
Clik here to view. $\Box$

This proof is tantalizingly similar to the proof of Frobenius’ lemma above; in both cases the key is to write down an action of Image may be NSFW.
Clik here to view. on a set of size relatively prime to Image may be NSFW.
Clik here to view. but such that all stabilizers must be Image may be NSFW.
Clik here to view.-groups. But I’m not yet sure how to relate them.

We’re not done with proofs of Sylow I! The next proof is due to Frobenius; Waterhouse stresses that its early use of quotients crucially highlighted the importance of working with abstract groups rather than just groups of substitutions as was standard in early group theory. It generalizes the class equation proof of Cauchy’s theorem above.

Proof 4 (class equation). We again induct on the order of Image may be NSFW.
Clik here to view. |G| and start by observing that Sylow subgroups clearly exist for finite abelian groups, for example by the structure theorem. We can also use the Chinese remainder theorem.

Now, suppose that Image may be NSFW.
Clik here to view. $p \mid |G|$ . If Image may be NSFW.
Clik here to view. also divides the order Image may be NSFW.
Clik here to view. |Z(G)| of the center, then Image may be NSFW.
Clik here to view. Z(G) contains a (central, abelian, nontrivial) Sylow Image may be NSFW.
Clik here to view.-subgroup Image may be NSFW.
Clik here to view., which we can quotient by. Then Image may be NSFW.
Clik here to view. G/A , by the inductive hypothesis, also contains a Sylow Image may be NSFW.
Clik here to view.-subgroup, and the preimage of this subgroup is a Sylow Image may be NSFW.
Clik here to view.-subgroup of Image may be NSFW.
Clik here to view..

If Image may be NSFW.
Clik here to view. $p \nmid |Z(G)|$ , then again we inspect the class equation

Image may be NSFW.
Clik here to view. $\displaystyle |G| = |Z(G)| + \sum_i \frac{|G|}{|C_G(g_i)|}$

where again the sum runs over non-central conjugacy classes. Since Image may be NSFW.
Clik here to view. $p \mid |G|$ we conclude that there is some conjugacy class Image may be NSFW.
Clik here to view. [g_i] such that Image may be NSFW.
Clik here to view. $p \nmid \frac{|G|}{|C_G(g_i)|}$ , hence such that Image may be NSFW.
Clik here to view. |G| and Image may be NSFW.
Clik here to view. have the same Image may be NSFW.
Clik here to view.-adic valuation. By the inductive hypothesis, Image may be NSFW.
Clik here to view. contains a Sylow Image may be NSFW.
Clik here to view.-subgroup, which is then (by virtue of having the appropriate order) also a Sylow Image may be NSFW.
Clik here to view.-subgroup of Image may be NSFW.
Clik here to view.. Image may be NSFW.
Clik here to view. $\Box$

This proof is the first one we’ve seen that provides a halfway reasonable algorithm for finding Sylow subgroups, which is lovely (the first three are more or less just raw existence proofs): pass to the quotient by central Image may be NSFW.
Clik here to view.-groups until you can’t anymore, then search for conjugacy classes whose centralizers have index relatively prime to Image may be NSFW.
Clik here to view. and pass to the centralizers until you can’t anymore, then repeat.

We give one final proof. According to Waterhouse, this one is essentially Sylow’s original proof, and it naturally proves an apparently-slightly-stronger result, interpolating between Cauchy’s theorem and Sylow I. It also provides a (different) algorithm for finding Sylow subgroups.

Sylow I+: Every finite group Image may be NSFW.
Clik here to view. contains a subgroup of prime power order Image may be NSFW.
Clik here to view. p^k whenever Image may be NSFW.
Clik here to view. $p^k \mid |G|$ . These subgroups can be constructed inductively by starting from an element of order Image may be NSFW.
Clik here to view. and then finding an element of Image may be NSFW.
Clik here to view.-power order normalizing the previous subgroup.

(Note that this result is equivalent to Sylow I as usually stated once we know that finite Image may be NSFW.
Clik here to view.-groups admit composition series in which each quotient is Image may be NSFW.
Clik here to view. C_p , which is not hard to prove by induction once we know that finite Image may be NSFW.
Clik here to view.-groups have nontrivial center, which in turn we proved previously using the PGFPT. The below proof actually gives an independent proof of the existence of such a composition series, and hence an independent proof that finite Image may be NSFW.
Clik here to view.-groups are nilpotent.)

Proof 5 (normalizers + PGFPT). We induct on Image may be NSFW.
Clik here to view.. The base case Image may be NSFW.
Clik here to view. k = 1 is Cauchy’s theorem, which we proved above (four times!).

Now suppose we’ve found a subgroup Image may be NSFW.
Clik here to view. of order Image may be NSFW.
Clik here to view. p^k , and we’d like to see if we can find a subgroup of order Image may be NSFW.
Clik here to view. $p^{k+1}$ . Consider the action of Image may be NSFW.
Clik here to view. by left multiplication on the cosets Image may be NSFW.
Clik here to view. G/P . By the PGFPT the number of fixed points of this action satisfies

Image may be NSFW.
Clik here to view. $\displaystyle |(G/P)^P| \equiv |G/P| \bmod p$ .

On the other hand, the fixed points of this action are precisely the cosets Image may be NSFW.
Clik here to view. such that Image may be NSFW.
Clik here to view. PgP = gP , or equivalently such that Image may be NSFW.
Clik here to view. $g^{-1} P g = P$ . So they are naturally in bijection with the quotient Image may be NSFW.
Clik here to view. N_G(P)/P of the normalizer

Image may be NSFW.
Clik here to view. $\displaystyle N_G(P) = \{ g \in G : gPg^{-1} = P \}$

of Image may be NSFW.
Clik here to view.. So the PGFPT tells us:

Lemma: If Image may be NSFW.
Clik here to view. is a Image may be NSFW.
Clik here to view.-subgroup of a finite group Image may be NSFW.
Clik here to view., then Image may be NSFW.
Clik here to view. $|G/P| \equiv |N_G(P)/P| \bmod p$ .

If Image may be NSFW.
Clik here to view. $p^{k+1} \mid |G|$ , or equivalently if Image may be NSFW.
Clik here to view. is not a Sylow Image may be NSFW.
Clik here to view.-subgroup, then Image may be NSFW.
Clik here to view. $|G/P| \equiv 0 \bmod p$ and hence Image may be NSFW.
Clik here to view. $|N_G(P)/P| \equiv 0 \bmod p$ , so by Cauchy’s theorem it follows that Image may be NSFW.
Clik here to view. has an element Image may be NSFW.
Clik here to view. of order Image may be NSFW.
Clik here to view.. The preimage of the cyclic subgroup Image may be NSFW.
Clik here to view. $\langle g \rangle$ in Image may be NSFW.
Clik here to view. N_G(P) is then a Image may be NSFW.
Clik here to view.-subgroup of order Image may be NSFW.
Clik here to view. $p^{k+1}$ as desired; more specifically it’s a subgroup Image may be NSFW.
Clik here to view. given by some extension

Image may be NSFW.
Clik here to view. $\displaystyle 1 \to P \to P' \to C_p \to 1$

where Image may be NSFW.
Clik here to view. generates the copy of Image may be NSFW.
Clik here to view. C_p . So we’ve found our bigger Image may be NSFW.
Clik here to view.-subgroup. Image may be NSFW.
Clik here to view. $\Box$

This proof gives us an algorithm for finding Sylow subgroups by starting with elements of order Image may be NSFW.
Clik here to view. and repeatedly finding elements of order Image may be NSFW.
Clik here to view. p^k in the normalizer, and note that it also gives us a way to prove that a Image may be NSFW.
Clik here to view.-subgroup is Sylow without explicitly checking that its index is coprime to Image may be NSFW.
Clik here to view.:

Corollary (normalizer criterion): A Image may be NSFW.
Clik here to view.-subgroup Image may be NSFW.
Clik here to view. of a finite group Image may be NSFW.
Clik here to view. is a Sylow Image may be NSFW.
Clik here to view.-subgroup iff the quotient Image may be NSFW.
Clik here to view. N_G(P)/P contains no elements of order Image may be NSFW.
Clik here to view..

It has another corollary we haven’t quite gotten around to proving yet:

Corollary (Sylow = maximal): Sylow Image may be NSFW.
Clik here to view.-subgroups are precisely the maximal Image may be NSFW.
Clik here to view.-subgroups.

(Before it could have been the case that there were Image may be NSFW.
Clik here to view.-subgroups maximal with respect to inclusion but without index coprime to Image may be NSFW.
Clik here to view..)

These are all the proofs of Sylow I that I know or could track down. After having gone through all of them I think I like proof 4 (class equation) and proof 5 (normalizers) the best. Proof 3 (action on subsets) is really too opaque and too specialized; you only learn from it that Sylow subgroups exist and practically nothing else, whereas proof 4 and proof 5 both teach you more about them, including how to find them. I’m sort of amazed it took me over 10 years after I first ran into the Sylow theorems to finally see a version of Sylow’s original proof, and that when I did, I liked it so much better than the first proof I had seen.

Sylow II and III

These will be short.

Sylow II: Let Image may be NSFW.
Clik here to view. be a finite group and Image may be NSFW.
Clik here to view. be any Sylow Image may be NSFW.
Clik here to view.-subgroup. Every Image may be NSFW.
Clik here to view.-subgroup Image may be NSFW.
Clik here to view. of Image may be NSFW.
Clik here to view. is contained in a conjugate of Image may be NSFW.
Clik here to view.. In particular, taking Image may be NSFW.
Clik here to view. to be another Sylow Image may be NSFW.
Clik here to view.-subgroup, every pair of Sylow Image may be NSFW.
Clik here to view.-subgroups is conjugate.

Proof 1. Immediate corollary of Frobenius’ lemma. Image may be NSFW.
Clik here to view. $\Box$

Proof 2. Consider the action of Image may be NSFW.
Clik here to view. on Image may be NSFW.
Clik here to view. G/P . By hypothesis, Image may be NSFW.
Clik here to view. $|G/P| \not \equiv 0 \bmod p$ , so by the PGFPT,

Image may be NSFW.
Clik here to view. $\displaystyle |(G/P)^K| \equiv |G/P| \not \equiv 0 \bmod p$ .

A fixed point of the action of Image may be NSFW.
Clik here to view. on Image may be NSFW.
Clik here to view. G/P is precisely a coset Image may be NSFW.
Clik here to view. such that Image may be NSFW.
Clik here to view. KgP = gP , or equivalently such that Image may be NSFW.
Clik here to view. $g^{-1} K g \subseteq P$ , so there must be at least one such Image may be NSFW.
Clik here to view. conjugating Image may be NSFW.
Clik here to view. into Image may be NSFW.
Clik here to view. as desired. Image may be NSFW.
Clik here to view. $\Box$

Corollary (Sylow = maximal, again): Sylow Image may be NSFW.
Clik here to view.-subgroups are precisely the maximal Image may be NSFW.
Clik here to view.-subgroups.

(We’ve now proved this result in three different ways!)

Sylow III: Let Image may be NSFW.
Clik here to view. be a finite group and Image may be NSFW.
Clik here to view. be any Sylow Image may be NSFW.
Clik here to view.-subgroup. The number Image may be NSFW.
Clik here to view. n_p of Sylow Image may be NSFW.
Clik here to view.-subgroups of Image may be NSFW.
Clik here to view. satisfies Image may be NSFW.
Clik here to view. n_p = |G/N_G(P)| and Image may be NSFW.
Clik here to view. $n_p \equiv 1 \bmod p$ ; in particular, Image may be NSFW.
Clik here to view. $n_p \mid |G|$ .

Proof. Since the Sylow Image may be NSFW.
Clik here to view.-subgroups are conjugate, the set of Sylow Image may be NSFW.
Clik here to view.-subgroups is just the set of conjugates of Image may be NSFW.
Clik here to view.. Image may be NSFW.
Clik here to view. acts transitively on this set by conjugation with stabilizer Image may be NSFW.
Clik here to view. N_G(P) , so by orbit-stabilizer it has size Image may be NSFW.
Clik here to view. n_p = |G/N_G(P)| and in particular has size dividing Image may be NSFW.
Clik here to view. |G| . We now give two proofs of the all-important congruence Image may be NSFW.
Clik here to view. $n_p \equiv 1 \bmod p$ .

Subproof 1 (action of Image may be NSFW.
Clik here to view. on Image may be NSFW.
Clik here to view.). In Proof 5 (normalizers + PGFPT) of Sylow I above we showed that

Image may be NSFW.
Clik here to view. $\displaystyle |G/P| \equiv |N_G(P)/P| \not \equiv 0 \bmod p$

and dividing both sides by Image may be NSFW.
Clik here to view. $|N_G(P)/P| \bmod p$ gives the desired result.

Subproof 2 (action of Image may be NSFW.
Clik here to view. on Image may be NSFW.
Clik here to view.). This proof has the benefit of explaining what this congruence “means.” Image may be NSFW.
Clik here to view. N_G(P)/P can’t be divisible by Image may be NSFW.
Clik here to view., so the only elements of Image may be NSFW.
Clik here to view.-power order that normalize Image may be NSFW.
Clik here to view. are the elements of Image may be NSFW.
Clik here to view., and the same must be true for any other Sylow. This implies that if we restrict the conjugation action of Image may be NSFW.
Clik here to view. on the Sylows to Image may be NSFW.
Clik here to view., then Image may be NSFW.
Clik here to view. fixes only itself, so by the PGFPT

Image may be NSFW.
Clik here to view. $\displaystyle |(G/N_G(P))^P| = 1 \equiv |G/N_G(P)| = n_p \bmod p$

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

Meditation on the Sylow theorems I

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