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.
Subtitle: “The size of the normaliser of a maximal -subgroup has coprime to ”
There was a -subgroup of
(by Cauchy). The largest was .
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 :
is in ,
so applies. From this, we’ll see:
Subtitle: “ is the only orbit of size ”
Call it . If false,
a hom 3.
(Lagrange). Then Lemmarick proves:
Trochaic Tetrameter Tying Together 4
Subtitle: “ is Sylow, since has size coprime to ”
has kernel - but also
is , so lies inside it.
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
possibly divide those two. So
Maximal the power of is:
’s a Sylow -subgroup.
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:
The previous poem’s our saviour:
is in .
The Pigeonhole tells its behaviour:
that is .
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. ↩
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 “sum-A”. ↩