--- lastmod: "2022-01-01T22:20:19.0000000+00:00" author: patrick categories: - creative - mathematical_summary comments: true date: "2013-08-31T00:00:00Z" math: true aliases: - /wordpress/archives/379/index.html - /creative/mathematical_summary/slightly-silly-sylow-pseudo-sonnets/index.html - /slightly-silly-sylow-pseudo-sonnets/index.html title: Slightly silly Sylow pseudo-sonnets --- This is a collection of poems which together prove the [Sylow theorems][1]. # 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][4] Motivation [^notsonnet] 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][5] *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](https://en.wikipedia.org/wiki/Jerusalem_%28hymn%29)") *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][6] Claim [^cinquain] *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 [^hom]. Also, \\( \vert K \vert \\) divides \\( \vert N/P \vert \\) (Lagrange). Then Lemmarick proves: \\(K\\) Is \\({1}\\). ## [Trochaic Tetrameter][7] Tying Together [^rhyme] *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. # Theorem Two - Quad-[quatrain][8] [^quatrain] 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 [Haiku][9] [^haiku] \\( \vert X \vert \\): \\(1 \pmod{p}\\) Orbit \\(X\\) divides \\(G\\)'s size: We have proved the Third. [^notsonnet]: 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. [^cinquain]: I use a form of reverse cinquain, with syllable count 2,8,6,4,2,2,4,6,8,2. [^hom]: "Hom", of course, is short for "homomorphism". Imre Leader used it all the time, so I took it to be legitimate. [^rhyme]: 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. [^quatrain]: 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. [^haiku]: 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". [1]: {{< ref "2013-06-26-sylow-theorems" >}} [2]: http://tartarus.org/gareth/ [3]: http://mmeblair.tumblr.com/post/61532912275/carnival-of-mathematics-102-my-summation-of-other [4]: https://en.wikipedia.org/wiki/Monorhyme [5]: https://en.wikipedia.org/wiki/Limerick_%28poetry%29 [6]: https://en.wikipedia.org/wiki/Cinquain [7]: https://en.wikipedia.org/wiki/Trochaic_tetrameter [8]: https://en.wikipedia.org/wiki/Quatrain [9]: https://en.wikipedia.org/wiki/Haiku