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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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.