Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://nuclphys.sinp.msu.ru/mirrors/1996_10c.pdf Äàòà èçìåíåíèÿ: Tue Jan 13 12:39:05 2009 Äàòà èíäåêñèðîâàíèÿ: Tue Oct 2 00:14:28 2012 Êîäèðîâêà: Windows-1251

GROUP CALCULUS
A. Yu. OL'SHANSKII An axiomatic approach to the group definition, one of the most important in mathematics, is given. The definitions are accompanied by clear examples and exercises that are useful for optional classes at high school. A new proof of the Cauchy theorem is found not to be beyond the scope of the high-school mathematical programs. è ÂÒÚ,ÎÂÌ ÍÒËÓÏÚË~ÂÒÍËÈ ÔÓiÓ Í ÔÓÌflÚË ,, ÛÔÔœ - ÓÌÓÏÛ ËÁ ,ÊÌÂÈ?Ëi , ÏÚÂÏÚËÍÂ. è Ë,ÓËϜ ÓÔ ÂÂÎÂÌËfl ÒÓÔ Ó,ÓÊÚÒfl Ô ÓÒÚœÏË Ô ËÏ ÏË Ë ÛÔ ÊÌÂÌËflÏË, ÔÓÎÂÁÌœÏË Îfl ÙÍÛÎåÚÚË,Ìœi ÁÌflÚËÈ , ?ÍÓÎÂ. óÚÓžœ ÒÂÎÚå ÏÚ ËÎ Ì ,œiÓflËÏ Á ÏÍË ?ÍÓÎåÌÓÈ Ô Ó,, ÏÏœ, Îfl ÚÂÓ ÂÏœ äÓ?Ë ÌÈÂÌÓ ÎÂÏÂÌÚ ÌÓ ÓÍÁÚÂÎåÒÚ,Ó.


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

1. ÇÇÖÑÖçàÖ

. [1] . . , . . , 2 , . - .
2. èêéàáÇÖÑÖçàÖ ëàååÖíêàâ

. , (. 1) ? . -, , . -, AB , 90œ, 180œ 270œ. ( .) -, 1-7 120œ 240œ. ( : 2-8, 3-5 4-6.) , CD, ,
2 A 1 C 4 3

¿ éÎå?ÌÒÍËÈ Ä.û., 1996

6 B 5 . 1.

D

7

8

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180œ. ( .) , , 1 + 3 × × 3 + 2 4 + 6 = 24 ( 24). , - , AB 90œ ( 1 2, 2 3 ..), 120œ 1-7 ( 2 4, 4 5 ..)? . , 1 2, 2 4. , 1 4. 3 5, 2 8, 4 6 . , CD 180œ. , , . "", , . h = fg - f g - - X (, ) x y ( , x ) y = h(x) = f (g(x)): x g, - f. x y h. 1 2, (1 ) = 2. (2 ) = 4, (1 ) = ((1 )) = (2 ) = 4 = . ( )( 1 ) = = ((1 )) = (1 ) = 2 ( (1 ) = 1 ), , , . , ( ) . , f, g, h (fg)h = f (gh) ( , , ). ((fg)h)(x) = (fg)(h(x)) = f (g(h(x))) = f ((gh))(x)) = = (f (gh))(x) x. - X e, X : e(x) = x. , : ef = fe = f -

f, e (f (x)) = f (x) f (e (x)) = f (x) . , f g fg = gf = e. g = f -1. g g(y) = x, f (x) = y. f: y X , , x. -1 2 1, 1 - 4, 6 - 5 ..
3. éÅôÖÖ èéçüíàÖ Éêìèèõ

. () G, - "". ( [1] , G - .) G, . , a b G, , d, ab. G , : ) : (ab)c = a(bc) a, b, c G; ) G ( "") e, , ae = ea = a a G; ) a G b, , ab = ba = e. , , , . , , (, ), . , 24. , , ) - ) . ) . , - e', ee' = e' ( e ) ee' = e ( e' ). e' = e. a. b b' - , ) - ) b = be = b(ab') = (ba)b' = eb' = b',

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115


b = b'. : b = a -1. . ac = ad , c = d. ac, ad, a -1. a -1ac = a -1ad, ec = ed c = d. . , ca = da c = d. ca = da a -1. G x ax = b (1)

(x0 , y0 , z0) xyz = b x0 y0 , az = b, a = x0y0 , z0 . (x0 , y0), x0 , y0 G, n2, n - G. ( n × n, x0 , - y0 .) , xyz = b n n2 . x1x2...xn = b x1 , x2 , ..., xp np - 1 G.
4. ëÇüáú ë ÉêìèèÄåà èêÖéÅêÄáéÇÄçàâ

a, b G. (1) a -1: a -1ax = a -1b, ex = a-1b x = a -1b. a -1b (1) x. , (1) a = , b = , , -1. , x(1 ) = ( -1)(1 ) = -1(1 ) = 4. x(2 ) .., , x 90œ , 1-4-8-5 2-3-7-6. xa = b x = ba-1, 90œ , 1-2-6-5 4-3-7-8, 2 1 .. (!) axb = c, xa = b ( ), a - 90œ, b - . G n : xy = b ? (2)

, "" . , , , . , G , ) - ) . ( e = 1 - .) ) - ) , . G , 0, . G . , k 0 : Fk - G, x G k ; , Fk(x) = kx. , k = 3 , F3(x) = 3x, F3 - 3 . , : FkFl = Fkl , Fl Fl x G kl, Fkl . , G. , G , Fk , k G. , . [1], .

, (x0 , y0) x0 y0 G, (2) x0y0 = b. x0 G , (2) x0y = b y, , -1 , y = x 0 b . x0 n G (2) n n.

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5. íÄÅãàñõ ìåçéÜÖçàü

, , . ( G) , . 2. , AB CD - a b , 180œ - c. ab (, b, a), C

A

B

D
. 2.



, ab = c. ac = b, aa = e .. (. 1) , a, , b, c . . . 2 ( H) 0œ ( e), 90œ ( a), 180œ (b) 270œ (c) 4 × 4 (. 1), ac = e, aa = b .., . 1 1 . , ( a 90œ, 180œ . .). : , 1 e a e a b c e a b c a e c b

, , - . " G H ". , 4 : , , ( 3 3 ..). , ... , , 1, - H2O. , , 2 ( ) 3. 4 , . 1. , , . (. . 1 4). , , . , , .
6. èéêüÑäà ëàååÖíêàâ à ùãÖåÖçíéÇ Éêìèèõ

b b c e a

c c b a e e a b c

e e a b c

a a b c e

b b c e a

c c e a b

. 120œ , . , fff f e, f 3 = e. f n ( f n) n- f . 90œ 4- , 180œ - 2- . , . , 8 , 8 4 2, . ,

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, , , , . a G , , a0 = e : a0 = e, a1 = a, a2, a3, ... (3) : a -1 - a , , , a -2 = a -1a -1 = (a -1)2, ..., a -m = = (a -1)m m. a ka l = a k + l (a k )l = a kl ( ) k l. G (3) , k l a k = a l. k > l. a l a k = ea l a k - l = e, k - l > 0. a n, , a n = e. n - a, n e = a0, a, a2, ..., an
-1

) () 24 ( 48), . 1. n a G. - b, (4), n "a-": b = be, ba, ba2, ..., ban - 1. (5) , bak = ba l b 6 ak = a l. , G - c, (4) (5). a-: c, ca, ca2, ..., can - 1, (6) , (6), n . , (5) (6) . ba s = ca t - s t. a- t , c = ba s - t. a, a s - t, (4) (. 6). c = ba s - t , c (5) c. (4) (5) (4) (6). (4) - (6) G, , d G, , d, da, da2, ..., da n - 1 .. G , n . , G n 1 . , , , p G G p. , 24 6, 8 12, 4 (. 1) 4. ( .) p 3 . , p = 2 n G p = 2, n . G : xy = e. (7) (2) b = e, , n . x0y0 = e, -1 -1 x 0 , x0 , x 0 x0y0x0 = -1 = x 0 ex0 ,

(4)

, a k = a l , 0 l < k n - 1 , , a k - l = e, 0 < k - l < n, n a n = e. am (4), an = e, an + 1 = = an a = ea = a, ... a-1 = an - 1, an - 1a = aan - 1 = = an = e, a-2 = an - 2 .. , a n , n - a.
7. ÑÇÄ ëÇéâëíÇÄ èéêüÑäéÇ

1. G G. - . , , . , , ( -

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ëéêéëéÇëäàâ éÅêÄáéÇÄíÖãúçõâ ÜìêçÄã, <10, 1996


y0x0 = e. (y0 , x0) - (7), (y0 , x0) . x0 y0 . , , , . n (7) , (x0 , x0) , N x0 G 2 x 0 = e. , , : e2 = e. N > 0, N , N > 1. , 2 x0 e, , x 0 = e. , 2. p = 3, n 3. xyz = e n2 (. 3), , n, 3. (x0 , y0 , z0) (y0 , z0 , x0) (z0 , x0 , y0), , p = 2. - , (x0 , y0 , z0) = = (y0 , z0 , x0), x0 = y0 , y0 = z0 , z0 = x0 , , 3 x 0 = e . , ( ) , , , 3 3 x0 G x 0 = e . , x0 . 2 , , x 0 = e, 3 x 0 = e , x0 = e.

, 3, 3. , p - , , 1 p . 2. p - n G, G p. p = 2, 3. , x1x2...xp = e c x1 , x2 , ..., xp , n p - 1 (. 3). p? , p = 5. p = 4 ?
ãàíÖêÄíìêÄ
1. .. // . 1996. 5. . 115 - 120.

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

éãúòÄçëäàâ Ä.û. ÉêìèèéÇõÖ àëóàëãÖçàü

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