Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://nuclphys.sinp.msu.ru/mirrors/1996_5c.pdf Äàòà èçìåíåíèÿ: Tue Jan 13 12:39:05 2009 Äàòà èíäåêñèðîâàíèÿ: Tue Oct 2 00:06:27 2012 Êîäèðîâêà: Windows-1251

MULTIPLICATION OF SYMMETRIES AND TRANSFORMATIONS
A. Yu. OL'SHANSKII This paper introduces the group theoretical point of view of symmetries and transformations. Symmetries of polynomials are used as a definition for permutation's sign, which is applied afterwards to a puzzle. Multiplication of symmetries is quite different from usual numerical operations. Therefore a "calculus" arises which leads to general notions of the group theory. ëÚÚåfl ÁÌÍÓÏËÚ Ò ÚÂÓÂÚËÍÓ-,, ÛÔÔÓ,ÓÈ ÚÓ~ÍÓÈ Á ÂÌËfl Ì ÒËÏÏÂÚËË Ë Ô ÂÓž ÁÓ,ÌËfl. ë ÔÓÏÓå ÒËÏÏÂÚ ËÈ ÏÌÓ,,Ó~ÎÂÌÓ, ,,ÓËÚÒfl ÔÓÌflÚË ~ÂÚÌÓÒÚË Ô ÂÒÚÌÓ,ÍË, ËÒÔÓÎåÁÛÂÏÓ Îfl ÓžflÒÌÂÌËfl ÓÌÓÈ ËÁ,ÂÒÚÌÓÈ Ë,, œ. ìÏÌÓÊÂÌË ÒËÏÏÂÚ ËÈ ÒÓ,ÒÂÏ Ì ÔÓiÓÊÂ Ì Ô Ë,œ~̜ ~ËÒÎÓ,œÂ ÓÔÂ^ËË. ÇÓÁÌËÍÂÚ ËÒ~ËÒÎÂÌËÂ, ÍÓÚÓ Ó , ÒÎÂÛÂÈ ÒÚÚåÂ Ô Ë,ÂÂÚ Í ÓžËÏ ÔÓÌflÚËflÏ ÚÂÓ ËË ,, ÛÔÔ.


Ä. û. éãúòÄçëäàâ
åÓÒÍÓ,ÒÍËÈ ,,ÓÒÛ ÒÚ,ÂÌÌœÈ ÛÌË, ÒËÚÂÚ ËÏ. å.Ç. ãÓÏÓÌÓÒÓ,

1. ÇÇÖÑÖçàÖ

, , , . , , , , , , . , , "" . ( "") , , , , . , , - , . . - . , " ", .
2. ëàååÖíêàà

¿ éÎå?ÌÒÍËÈ Ä.û., 1996

. , , , . , AB CD (. 1), 1 - 3 2 - 4

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C 1 A 4 D 3 2 BA

C B



D

. 1.

0œ, 90œ, 180œ 270œ . 8 "" 0œ ( , ). , , . 360œ , 90œ 450œ: , . (. 1) 3 : 180œ AB CD. (. 1) . , . ( ) , , (. 2) ( ), (. 2), 3 a, 3 b, 6 a, 6 b, ... ka + l b , k - l k l 3. xy + xz + yz , xy 2 + yz 2 + zx 2,

, x + 2y + 5z. , "" , , x, y z . , , , y, y z, z x. x 1. , 1 (!), , . 1 xyz yxz xyz yzx xyz zyx xyz zxy xyz xzy xyz xyz

, , .. , . , 6, 8. , , .
3. èêÖéÅêÄáéÇÄçàü

a

b

. 2.

- "" (, , ). f X ? X . ( "". !) , ( , , ) f X, a X (a X ) X, a f (a); , b X (b X ) , , a X. a b. .

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1. X - , f - 90œ . (1, 0) (0, - 1), (0, - 1) (- 1, 0). , (x, y) (y, - x), (x, y) (- y, x). 2. X - , f - n (1 , 2). A (a1 , a2) (a1 + 1 , a2 + 2), - (a1 - 1 , a2 - 2). 3. X - . y = f (x) X, y x f (x) = 2x - 6 f (x) = x 3. f (x) = x2 , , , 4 ( 4 ) : 2 - 2, - 1 . 4. - X, , "". , , X x, y, z. f, f (x) = y, f (y) = z f (z) = x, 1, X 6, . X, n , . 1 n. ( , , x 1, y - 2, z - 3.) f ( X ) : X ( , - 1, ..., n), f. , f f = 123 , 231 1 2. 2 f 1 = 123 , 213 f 4 = 123 , 231 f 2 = 123 , 321 f 5 = 123 , 312 f 3 = 123 , 132 f 6 = 123 . 123

. : (1) n = 1, : (1, 2) (2, 1) n = 2 : (2, 1, 3), (3, 2, 1), (1, 3, 2,), (2, 3, 1), (3, 1, 2), (1, 2, 3) n = 3. , n X, n , 1 2 3 ... (n - 1) n = = n!. ( "n ".) , n. , , 1 n, n + 1 , 1 n + 1, n + 1 : , , ..., n-. , 1 n + 1, n + 1 , , 1 n, , n!(n + 1) = (n + 1)!.
4. èêéàáÇÖÑÖçàÖ èêÖéÅêÄáéÇÄçàâ

"" , , , , . f g - - X. fg, h, : x X h(x) = f (g(x)). , h(x) y = g(x), z = f (y) y f. , , h - X. . 1. f f (x) = 2x + 6 x. g(x) = x3. h = fg , h(x) = f (g(x)) = f (x 3) = 2x 3 + 6, x x3 , f. , h' = gf h'(x) = g(f (x)) = g(2x + 6) = (2x + 6)3. , , fg gf. 2. - 2, f1 f2 . X : X = {1, 2, 3}. ,

, f X, f

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h = f1 f2 h(1) = f1( f2(1)). 2 , f2(1) = 3, f1(3) = 3, h(1) = 3. : h(2) = 1 h(3) = 2. , f1 f2 , f5 2, , f1 f2 = f5 . , f2 f1 = f4 , f1 f2 , , , f4 f5 : f4 f5 = f5 f4 . 3. , , , 2 - 4 AB (. 1) 90œ , - 270œ . . - : , ..
5. Éêìèèõ èêÖéÅêÄáéÇÄçàâ

, fgh, . , e. x : e(x) = x, , e "" ( ) : ef = fe = f f, e(f (x)) = f (x) f (e(x)) = f (x) e. 2 e = f6 . f g : fg = gf = e; g = f - 1. f - 1, f -1(b) = a, f (a) = b, f f - 1. , (f - 1)- 1 = f. , 2 f
-1 1

= f 1,

f

-1 2

= f 2,

f

-1 3

= f 3,

f

-1 4

= f 5,

f

-1 6

= f 6.

. -, - X. X. , , , ( "" , , "" , , , ). , , . -, , - . - : , , f g , fg, , , g f. , . , ( ) . , ( ) f, g, h X (fg)h = f (gh) ( ). , , , , , x X f (g(h(x))).

f (x) = x 3 f (x) = 2x + 6 x g ( x ) = 3 x g ( x ) = -- - 3 . (2 !) , f ( ), f - 1 - .
6. óÖíçéëíú èÖêÖëíÄçéÇéä

2, n x1 , x2 , ..., xn , n - . Sn 1, 2, ..., n, n! (. 3). Sn n . , f 1 - i1 , 1 2...n 2 - i2 , ..., n - in ( f = ), i1 i2 ... in x1 xi1 , x2 - xi2, ..., xn - xin. , x1 - 2x2 + 5x3x4 , 1234 , x2 - 2x4 + 5x1x3 . 2413 n = 3 2 2 2 x1 + x2 + x3 (x1 + x2)(x1 + + x3)(x2 + x3) ( ) , . d3(x1 , x2 , x3) = (x1 - x2)(x1 - x3)(x2 - x3)? , 2,

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, . (!) d3(x1 , x2 , x3) , . d3(x1 , x2 , x3) , . , f4 , f5 f6 2. , f1 , f2 , f3 - . 1 2...n f = i1 i2 ... in 1 n, n x1 , x2 , ..., xn : dn(x1 , x2 , ..., xn ) = (x1 - x2)(x1 - x3) ... (x1 - xn )(x2 - x3) ... (1) ... (x2 - xn )(x3 - x4) ... (x3 - xn) ... (xn - 1 - xn ). xk - xl dn(x1 , ..., xn) , k < l. f xk - xl xik - xil , (1), , (1) ( k < l, ik > il). , (1) + dn(x1 , ..., xn ). Sn , dn(x1 , ..., xn ), (1), , . , - k k + 1 1, ..., n: f = 12... k - 1 k k + 1 k + 2 ... n . 12... k - 1 k + 1 k k + 2 ... n (2)

. , , fg , g, f. , ( ) , g (1), f dn(x1 , ..., xn) - dn(x1 , ..., xn ), fg ( g, f ) dn(x1 , ..., xn ) - dn(x1 , ..., xn). . : "". , , ( ) . , , ( , ) k l : 12... k k + 1 ... l - 1 l ... n . 12... l k + 1 ... l - 1 k ... n (3)

, k < l, , k + 1 < l, (3) , , . g, k + 1 l : g = 12... k k + 1 k + 2 ... l - 1 l l + 1 ... n . 12... k l k + 2 ... l - 1 k + 1 l + 1 ... n g f, (2), gfg. , , gfg (3). (, k g , f k + 1, g l. k l, (3), . .) , f - . gfg , (3) . 1. . Sn n > 1 . , f1 , ..., fs - Sn , g1 , ..., gt - .

, (2), xk - xk + 1 xk + 1 - xk = - (xk - xk + 1). (2) , . , k + 1 < l, xk + 1 - xl xk - xl, - k < l, .. (2) (1), .
7. èêÄÇàãé óÖíçéëíà Ñãü èêéàáÇÖÑÖçàü èÖêÖëíÄçéÇéä

, , , -

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- h ( , n > 1). hf1 , ..., hfs ( hfi = hfj h- 1 fi = fj ). 1 s , , g1 , ..., gt . , s t. , hg1 , ..., hgt , : t s. , s = t, 3 , s + t = n!. , 2. n > 1 Sn n! /2.
8. àÉêÄ Ç "èüíçÄÑñÄíú" à èÖêÖëíÄçéÇäà

, 3 f = 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 , 3 14 6 8 5 2 16 11 13 4 15 10 9 7 1 12 3 - , 3 - , 14 15. , , . , 3 6 ( 16), f 6 16 . , f g, 6 16. ( , gf .) 2 ( 11 15) f , 2 16 (11 16 15 16 ). 1 , , , . , 3 3, . 3 3 , . , 3 3 "" 16 . , - , 3 3, , , , . , 3 3 . 3 3 (, , !). , (, , " ") , :
.., .. . .: , 1985.

, . "". , 1 15. (. 3). , , , (, , , ). 3, , 2 (), 11 (), 6 () 15 (). , , , 3 3. , 16, (, , ). 1 2 ... 16 f = , i1 i2 ... i16 i1 - , , i2 - , ..., i5 - . .
3 5 13 9 14 2 4 7 6 7 15 1 8 11 10 12 1 5 9 2 6 3 7 4 8 12 1 5 9 2 6 3 7 4 8 12 16

10 11

10 11

13 15 14 . 3.

13 14 15

*** , - , - . .. , . 50 , .

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