Slightly silly Sylow pseudosonnets
This is a collection of poems which together prove the Sylow theorems.
Notes on pronunciation
 Pronounce \( \vert P \vert \) as “mod P”, \(a/b\) or \(\dfrac{a}{b}\) as “a on b”, and \(=\) as “equals”.
 \(a^b\) for positive integer \(b\) is pronounced “a to the b”.
 \(g^{1}\) is pronounced “gee inverse”.
 “Sylow” is pronounced “seelov”, for the purposes of these poems.
 \(p\) and \(P\) and \(n_p\) are different entities, so they’re allowed to rhyme.
Monorhymic Motivation ^{1}
Suppose we have a finite group called \(G\).
This group has size \(m\) times a power of \(p\).
We choose \(m\) to have coprimality:
the power of \(p\)’s the biggest we can see.
Then One: a subgroup of that size do we
assert exists. And Two: such subgroups be
all conjugate. And \(m\)’s nought mod \(n_p\),
while \(n_p = 1 \pmod{p}\); that’s Three.
Theorem One
Little Lemmarick
Subtitle: “The size of the normaliser \(N\) of a maximal \(p\)subgroup \(P\) has \(N/P\) coprime to \(p\)"
There was a \(p\)subgroup of \(G\)
(by Cauchy). The largest was \(P\).
Let \(N\) normalise,
Take \(\dfrac{N}{P}\)’s size,
Suppose that it’s zero mod \(p\).
Now \(\dfrac{N}{P}\) also has some
psubgroup (by Cauchy); take one.
Take it unprojected,
\(P\)’s most big? Corrected!
We’ve found one sized \(p \vert P \vert \): done.
Introductory Interlude (to the tune of “Jerusalem")
Subtitle: “\({P}\) is an orbit of size \(1\) under the conjugation action of \(P\) on the set of \(G\)conjugates of \(P\)"
Let \(X\) be \(P\)’s orbit under \(G\)
Acting by conjugation.
Mod \(G\) o’er \(N\)’s the size of \(X\)
The Orbit/Stabiliser’s done.
And in its turn, \(P\) acts on \(X\)
By conjugating, as before,
Then \(P\) is certainly all alone:
Its orbit is itself, no more.
Let \(gPg^{1}\) be alone,
\(P\) stabilises it, and hence
\(pgPg^{1}p^{1}\)
Is \(gPg^{1}\)  from whence
We conjugate by \(g^{1}\):
\(g^{1}Pg\) fixes \(P\).
\(g^{1}Pg\) is in \(N\),
so \(\pi\) applies. From this, we’ll see:
Cinquain Claim ^{2}
Subtitle: “\({P}\) is the only orbit of size \(1\)"
A claim:
\(\pi(g^{1}Pg)\) is \({1}\).
Call it \(K\). If false, \(p\)
divides \( \vert K \vert \),
as \(\pi\)
a hom ^{3}.
Also, \( \vert K \vert \)
divides \( \vert N/P \vert \)
(Lagrange). Then Lemmarick proves: \(K\)
Is \({1}\).
Trochaic Tetrameter Tying Together ^{4}
Subtitle: “\({P}\) is Sylow, since \(G/N\) has size coprime to \(p\)"
\(\pi\) has kernel \(P\)  but also
\(K\) is \({1}\), so lies inside it.
\(P\) contains \(g^{1}Pg\);
Both have size \(p^a\). So
since they’re finite, they’re the same set.
Any set alone in orbit
must be \(P\). The class equation
Tells us \( \vert G \vert / \vert N \vert \) is
Just precisely \(1 \pmod{p}\). Then
\( \vert G \vert / \vert P \vert \) is not a
multiple of \(p\) because it’s
\( \vert \dfrac{N}{P} \vert \) multiplied by
\(\dfrac{ \vert G \vert }{ \vert N \vert }\) and \(p\) can’t
possibly divide those two. So
Maximal the power of \(p\) is:
\(P\)’s a Sylow \(p\)subgroup.
Theorem Two  Quadquatrain ^{5}
A Sylow \(p\)subgroup let \(Q\) be:
a subgroup, size \(p^a\).
Because it’s the same size as was \(P\),
it acts on \(X\) in the same way.
Mod \(p\), we have \( \vert X \vert \) is \(1\) 
the orbits of \(Q\) will divide it;
Now invoke the class equation:
an orbit, size \(1\), lies inside it.
We dub this one \(gPg^{1}\),
then \(g^{1}Qg\)’s in \(N\).
Projection works just as well in verse:
\(\pi(g^{1}Qg)\) is \({1}\).
The previous poem’s our saviour:
\(g^{1}Qg\) is in \(P\).
The Pigeonhole tells its behaviour:
that \(P\) is \(g^{1}Qg\).
Theorem Three  Hindmost Haiku ^{6}
\( \vert X \vert \): \(1 \pmod{p}\)
Orbit \(X\) divides \(G\)’s size:
We have proved the Third.

This is not a sonnet  it is six lines too short, and is monorhymic rather than following a more varied rhyme scheme. I started out intending it to be a sonnet, but all the rhymes for “p”, “G” and so forth were irresistible. “Power” is a monosyllable. ↩︎

I use a form of reverse cinquain, with syllable count 2,8,6,4,2,2,4,6,8,2. ↩︎

“Hom”, of course, is short for “homomorphism”. Imre Leader used it all the time, so I took it to be legitimate. ↩︎

This section is unrhymed; although Shakespeare rhymes his tetrameter, Longfellow doesn’t. The strong iambic nature of English makes enjambement very natural to write when you’re constrained to trochees, so I have just gone with the flow. ↩︎

Quatrains have a variety of allowable rhyme schemes, but I plumped for ABAB for the sake of variety. Yes, “N” rhymes with “one”. For the purposes of scansion, pronounce each line as the first line of a limerick, with an optional weak syllable at the end if necessary. ↩︎

I know that a haiku should mention a season, etc  but that is a constraint I am willing to relax. Gareth pointed out that if “sum” and “size” were synonymous, then " X : 1 (mod p)/Orbit X divides G’s sum/A proof of the Third” would mention the season “sumA”. ↩︎