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Patrick Stevens

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

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This is a collection of poems which together prove the Sylow theorems.

Notes on pronunciation

  • Pronounce as “mod P”, or as “a on b”, and as “equals”.
  • for positive integer is pronounced “a to the b”.
  • is pronounced “gee inverse”.
  • “Sylow” is pronounced “see-lov”, for the purposes of these poems.
  • and and are different entities, so they’re allowed to rhyme.

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 we can see.
Then One: a subgroup of that size do we
assert exists. And Two: such subgroups be
all conjugate. And ’s nought mod ,
while ; that’s Three.

Theorem One

Little Lemmarick

Subtitle: “The size of the normaliser of a maximal -subgroup has coprime to

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
p-subgroup (by Cauchy); take one.
Take it un-projected,
’s most big? Corrected!
We’ve found one sized : done.

Introductory Interlude (to the tune of “Jerusalem”)

Subtitle: “ is an orbit of size under the conjugation action of on the set of -conjugates of

Let be ’s orbit under
Acting by conjuga-ti-on.
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. From this, we’ll see:

Cinquain Claim 2

Subtitle: “ is the only orbit of size

A claim:
is .
Call it . If false,
divides ,
a hom 3.
(Lagrange). Then Lemmarick proves:
Is .

Trochaic Tetrameter Tying Together 4

Subtitle: “ is Sylow, since has size coprime to

has kernel - but also
is , so lies inside it.
contains ;
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 - Quad-quatrain 5

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 .

The previous poem’s our saviour:
is in .
The Pigeonhole tells its behaviour:
that is .

Theorem Three - Hindmost Haiku 6

Orbit divides ’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”.