 ### 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”.