Vek zhivi vek uchis', ili chislo 6
Nov. 29th, 2002 12:54 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
posicOkazyvaetsya, gruppa perestanovok 6-elementnogo mnozhestva imeet vneshnij avtomorfizm. Ya prochital ob etom zdes':
Consider 6 things, e.g. the numerals 123456. There are 15 non-ordered pairs of them. (Call these "duads.") Also there are 15 ways to divide the original set into three duads, e.g. {(12}{34}{56}}. Call these "synthemes." Five synthemes can be chosen so as to contain each duad exactly once; and there turn out to be exactly six ways to make this choice. You can label these six sets ABCDEF. Then a permutation of 123456 induces a permutation of ABCDEF. A two-cycle such as (12) induces a product of three disjoint two-cycles such as (AB)(CD)(EF), so the map from one permutation to the other cannot be an inner automorphism.
Dovol'no zamyslovataya vse-taki konstrukciya, sopostavlyayuschaya 6-elementnomu mnozhestvu drugoe 6-elementnoe mnozhestvo. Ya esche znayu na etu temu gorazdo bolee prostuyu konstrukciyu, sopostavlyayuschuyu 4-elementnomu mnozhestvu 3-elementnoe mnozhestvo.
Ostal'nye gruppy Sn (pri n ne ravnom 6) vneshnih avtomorfizmov ne imeyut, chto netrudno dokazat', rassmotrev klass sopryazhennosti, sostoyaschij iz transpozicij, i ego obraz pri nashem avtomorfizme. Podrobnosti imeyutsya zdes' (fajl v formate postscript).
Consider 6 things, e.g. the numerals 123456. There are 15 non-ordered pairs of them. (Call these "duads.") Also there are 15 ways to divide the original set into three duads, e.g. {(12}{34}{56}}. Call these "synthemes." Five synthemes can be chosen so as to contain each duad exactly once; and there turn out to be exactly six ways to make this choice. You can label these six sets ABCDEF. Then a permutation of 123456 induces a permutation of ABCDEF. A two-cycle such as (12) induces a product of three disjoint two-cycles such as (AB)(CD)(EF), so the map from one permutation to the other cannot be an inner automorphism.
Dovol'no zamyslovataya vse-taki konstrukciya, sopostavlyayuschaya 6-elementnomu mnozhestvu drugoe 6-elementnoe mnozhestvo. Ya esche znayu na etu temu gorazdo bolee prostuyu konstrukciyu, sopostavlyayuschuyu 4-elementnomu mnozhestvu 3-elementnoe mnozhestvo.
Ostal'nye gruppy Sn (pri n ne ravnom 6) vneshnih avtomorfizmov ne imeyut, chto netrudno dokazat', rassmotrev klass sopryazhennosti, sostoyaschij iz transpozicij, i ego obraz pri nashem avtomorfizme. Podrobnosti imeyutsya zdes' (fajl v formate postscript).
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(а) ÐалÑа из PSL2(F9),
(б) ÑопÑÑжение неÑеÑной поÑÑановкой, и
(в) пÑоизведение двÑÑ Ð¿ÑедÑдÑÑÐ¸Ñ .
ÐÑи ÑÑом (а) пÑодлеваеÑÑÑ Ð´Ð¾ внеÑнего авÑомоÑÑизма S6, не ÑовÑем понÑÑно, как.
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Date: 2002-12-02 09:18 am (UTC)Esche interesnee mne bylo by uznat', pochemu A8 est' GL4(F2) ...
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Date: 2002-12-02 01:24 pm (UTC)no subject
Date: 2002-12-02 02:58 pm (UTC)0 -> Z -> Z[X4] + Z -> Z[X6] + Z -> Z[X3] -> 0,
gde X4, X6 i X3 sut' chetyreh-, shesti- i trehelementnye mnozhestva s ponyatno kakim dejstviem S4. I ya hochu drugih takih tochnyh chetverok znat', esli oni vdrug suschestvuyut.
Pochemu A5=PSL2(F5), ya mogu ob'yasnit'. Na samom dele, S5=PGL2(F5), i vot pochemu. Yasno, chto PGL2(F5) vkladyvaetsya v S6 -- shest' tochek proektivnoj pryamoj nad F5. Ya utverzhdayu, chto izvestnyj nam vneshnij avtomorfizm S6 perevodit etu podgruppu v stabilizator odnoj iz shesti tochek.
Kak my pomnim iz konstrukcii vneshnego avtomorfizma, eto znachit, chto gruppa PGL2(F5) dolzhna sohranyat' nekotoroe razbienie vseh 15 par razlichnyh tochek proektivnoj pryamoj na 5 troek (takoe chto kazhdaya trojka sostoit iz neperesekayuschihsya par). Vot eto razbienie: dve pary tochek vhodyat v odnu gruppu, esli ih dvojnoe otnoshenie ravno -1 (t.e. mozhno vybrat' koordinaty, v kotoryh pervaya para budet 0 i beskonechnost', a vtoraya 1 i -1).
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Date: 2003-06-12 10:42 pm (UTC)formu na V=F_5^2 ne predstavljajuschuju nul', det = 1, s tochnost'ju do umnozhenija na -1 (eto navernoe pochti to zhe, chto ty govorish' (?)).
Kstati, A_5 eto zhe esche gruppa ikosaedra, kak izvestno, ona zhe podgruppa v SL(2,C) tipa E_8.
Navernoe, trehmernoe predstavlenie PSL(2,F_5) mozhno podnyat' v char 0, potom uvidet', chto rezul'tat opredelen nad R, obraz ego -- gruppa
simmetrij ikosaedra, a pryamye vershin ikosaedra est' kakie-to "kanonicheskie" podnyatija obraza racional'nyh tochek
P^1(F_5) pri vlozhenii v proektivizaciju trehmernogo predtavslenija.. Vo vsyakom sluchae, stabilizator pryamoj soedinjajushej protivopolozhnye vershiny ikosaedra -- Borelevskaja podgruppa v PSL(2,F_5) (po-moemu).
5 predmetov kak-to mozhno cherez ikosaedr opredelit'. Chto li pokrasit' grani v 5 cvetov, tak chtoby vokrug vershiny byli vse 5 cvetov v opredelennom ciklicheskom porjadke, i togda gruppa simmetrij dejstvuet na mn-ve cvetov?
A ne znaet li kto realizacii grupp simmetrij 4-merhnyh platonovyh tel (H_4 i D_4(?)) kak grupp Chevalley?
Esche J-P Serre interesovalsya v kakie-to nedavnie gody special'nymi izomorfizmami konechnyh grupp Chevalley, tol'ko ja ne znaju, k chemu on prishel.
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Date: 2003-06-14 03:17 pm (UTC)Naprimer, A_6=PSL(2,F_9): ugadyvaem snachala pogruppu
U\subset A_6, sootv. obrazu strogo vernetreugol'nyh matric, t.e.
additivnoj gruppe F_9, a imenno U porozhdena dvumya nepereskeajushimisja 3-ciklami. Dalee usmatrivaem, chto normalizator N (oboznachim ego B) otozhdesvtljaetsja s Borelevskoj v PSL(2) (kljuchevoj moment -- dlya a\in F_9^* umnozhenie na a^2 sohranjaet paru pogrupp izomorfnyh Z_3 v additivnoj gruppe F_9). Dalee S_6/B otozhdestvyaetsya s P^1(F_9) t.k. N=F_9, i podgruppa soprjazhennaja s N dejstvuet
tranzitivno na S_6/B -- {1B}; i ubezhdaemsja, chto obraz
tochnogo deistvija S_6 na S_6/B soderzhit PSL(2,F_9) (a znachit s nim sovpadaet).
Izomorfizm S_5=PGL(2,F_5) mozhno stroit' tak zhe (no prosche);
a A_8=GL(4,F_2) -- navernoe tak zhe no slozhnee (nachinaem
s podruppy F_2^3 \subset A_8, kotoraya dolzhna perejti v matricy, otlichajuschiesja ot tozhdestvennoj v odnom stolbce,
prichem tol'ko vne diagonali).
Sorry, vse, navernoe, i tak razobralis', ili zabrosili..
No esli kakie-to esche takogo roda izomorfizmy simpatichnye
vsplyvut -- pishite.
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Date: 2003-10-18 11:34 pm (UTC)простой (и, видимо, стандартный) способ строить внешний автоморфизм А_6, а заодно доказывать, что PSL(2,F_5) = А_5 изложен в Алгебре Ленга (в последнем издании во всяком случае), упр 40-41 к главе 1
(+ замечание в конце главы 13).
Всем привет!
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