Vek zhivi vek uchis', ili chislo 6

[posic]

Okazyvaetsya, 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 S_n (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).

Comments

Wow

[(Anonymous)]

Now that's amazing! Well, i'm kind of young I suppose, but wow, i'm completely amazed. Forgive my outbursts, i'm a bored lost soul who stumbled upon your journal and read the english parts. I would have loved to read more, however, I am not fluent in russian in any shape or form, in fact, I barely know the slightest French. What a lesson it is to learn, I think I will concentrate much more on the ability to read other languages. Thank you for this experience.

It's always darkest before the Dawn.

-Dawnbringer