This is a collection of poems which together prove the Sylow theorems. Note that and and are different entities, so they’re allowed to rhyme. Pronounce as “mod P”, or as “a on b”, and as “equals”. is pronounced “gee inverse”, while for positive integer is pronounced “a to the b”. “Sylow” is pronounced “seelov”, for the purposes of these poems.
Some improvements were made by Gareth Taylor.
Update: this page has made the 102nd Carnival of Mathematics.
Monorhymic Motivation ^{1}
Suppose we have a finite group called .
This group has size times a power of .
We choose to have coprimality:
the power of ’s the biggest it can be.
Then One: a subgroup of that size do we
assert exists. Two: conjugate are Sy
low subgroups. And ’s nought mod
And ; that’s Three.
Theorem One
Little Lemmarick
There was a subgroup of
(by Cauchy). The largest was .
Let normalise,
Take ’s size,
Suppose that it’s zero mod .
Now also has some
psubgroup (by Cauchy); take one.
Take it unprojected,
’s most big? Corrected!
We’ve found one sized : done.
Introductory Interlude ^{2}
Let be ’s orbit when
Acts on it by conjugation.
Mod o’er ’s the size of
The Orbit/Stabiliser’s done.
And in its turn, acts on
By conjugating, as before,
Then is certainly all alone:
Its orbit is itself, no more.
Let be alone,
stabilises it, and hence
Is  from whence
We conjugate by :
fixes .
is in ,
so applies. That’s QED.
Cinquain Claim ^{3}
I claim:
is .
Call it . If false,
divides ,
as
a hom ^{4}.
Also,
divides
(Lagrange). Then Lemmarick proves:
Is .
Trochaic Tetrameter Tying Together ^{5}
has kernel  but also
is , so lies inside it.
is in ;
Both have size . So
since they’re finite, they’re the same set.
Any set alone in orbit
must be . The class equation
Tells us is
Just precisely . Then
is not a
multiple of because it’s
multiplied by
and can’t
possibly divide those two. So
Maximal the power of is:
’s a Sylow subgroup.
Theorem Two  Quadquatrain ^{6}
A Sylow subgroup let be:
a subgroup, size .
Because it’s the same size as was ,
it acts on in the same way.
Mod , we have is 
the orbits of will divide it;
Now invoke the class equation:
an orbit, size , lies inside it.
We dub this one ,
then ’s in .
Projection works just as well in verse:
is .
Continue the previous exercise:
is in .
They both have exactly the same size,
so is .
Theorem Three  Hindmost Haiku ^{7}
:
Orbit divides ’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. ↩

To be sung to the tune of Jerusalem. I loved the final rhyme too much to get rid of it despite its factual inaccuracy, but it doesn’t affect the proof. ↩

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. The metre of the final line is intentional. ↩

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. ↩

Yes, 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 a season. ↩