 ### Patrick Stevens

Former mathematics student at the University of Cambridge; now a software engineer.

# Slightly silly Sylow pseudo-sonnets

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 “see-lov”, 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
p-subgroup (by Cauchy); take one.
Take it un-projected,
$$P$$’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 conjuga-ti-on.
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.

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 Haiku6

$$\vert X \vert$$: $$1 \pmod{p}$$
Orbit $$X$$ divides $$G$$’s size:
We have proved the Third.

1. 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.

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

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

4. 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.

5. 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.

6. 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 “sum-A”.