Белоногов. Задачник по теории групп
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{i, j} {1, . . . , n}
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$ tij (α) α = 0 "
( 2 # $ GLn(K) $ tij (1) di(α) α = 1
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0 @ SLn(K) 4 $
" 2 # $ Diagn(K) GLn(K)
# > K E n > 1 5 %-
( GLn(K) = tij (α), d1(β) | α, β K, i = j ,
. SLn(K) = tij (α) | α K, i = j .
$ F = (M, +, ·) E ) M " $"
H a b = a + b + ab > (M \{−1}, ) E ! ,
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4 # , $ Zn $ Zm[a]m E " m! # a
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k
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( A B E $ > (a, b)!
a A b B! $ (a1, b1) + (a2, b2) = (a1 + a2, b1 + b2)
(a1, b1)(a2, b2) = (a1a2, b1b2) $ |
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iB : a → aB (a Aut (V ))
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char (F ) |G|
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. G ≤ GL(V )! V V (n, F ) I V G
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W V = U + W U ∩ W = {0}
V U V
!$ F E $ G E J ,
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! {x1, . . . , xm} ∩ {z1, . . . , zs} = I "
" 0 " (x1)(x2) . . . (xn)!
(xi) " i {1, . . . , n} I xg = x! x X
g SX ! ! x g
g SX g = g1 . . . gm E g
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g l1, . . . , lm E : $!
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g (r ≥ 0) (1, . . . , 1, l1, . . . , lm) |
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J # # :
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X g SX ) g $
: P :
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! = −1 " g "
sign(g) J 4 SX
# SX -./ ! AX
X > ! AX
# ∆(x1, . . . , xn) -&0
X = {1, . . . , n} An n = 1
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A -0/ Sn
! > X E
( =" $ SX
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#
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0 I g E (m1, . . . , ms) SX ! o(g) = H.O.K.(m1, . . . , E m1, . . . , ms
! ( A : S4 $
. $ S4.
! ) # : Sn
n ≤ 10
! ( 7 Sn (n − 1)! : n
. I a Sn o(a) ! o(a) ≤ n
! n N m = pa11 . . . pass ! pi E
ai N ; H
( Sn : m! . pa11 + . . . + pass ≤ n
! 3 : n Sn! n = 2p = 4
n = pm! p E m N
! ! a E $ m Sn! k E
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! 3 " SnH ( (i1i2 . . . im) = (i1i2)(i1i3) . . . (i1im)!
. (ij) = (ki)(kj)(ki) k {i, j}! 0 (ij)(ik) = (ijk) k = j!
% (ij)(kl) = (ijk)(ilk) {i, j} ∩ {k, l} = ! & (ijk)−1 = (kji)!
5 (ijk)−1(ijl)(ijk) = (jkl)!
- z = (1 2 . . . n)! z−1(ij)z = (i + 1 j + 1) i < n, j < n z−1(i n)z = (i + 1 1)
! " n ≥ 2
( Sn = (12), |
(13), . . . , (1n) ! |
. Sn = (12), |
(23), . . . , (n − 1 n) ! |
0 Sn = (12), |
(12 . . . n) = (12), (23 . . . n) |
! I a b E $ Sn, o(ab) ≤ 3.
! a = (a1, ..., ak) b = (b1, ..., bl) E $
(k, l N) A ab(a1, b1).
! z = (a1 . . . ak) 1 ≤ i < j ≤ k A H z(ai aj ), (ai aj )z (ai aj )z(ai aj )
! A " $ k, l, m N)H
( (a1, ..., ak, b1, ..., bl, c1, ..., cm)(a1, b1, c1).
. (a1, ..., ak, b1, ..., bl, c1, ..., cm)(a1, b1, c1)−1.
! ! ( # Sn
$
. < " # x! " # t ! "
$ s 0 ≤ s ≤ t $ z #! xz $
n − s
! " g, h Sn.
( sign(gh) = sign(g) sign(h).
. I h E $ ! g gh
0s I (z1, . . . , zs) E $# g! sign (g) = (−1)t! |
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t = =1(zi |
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% sign(g) = |
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! # 2 ≤ |X| < ∞
( SX
$# SX D
. Z4 SX 4
$# 0 2 4 4
$#
!$ N |X| = n N
( J 4 X
SX AX
. AX An
0 I n ≥ 2, |An| = n!/2 |Sn : An| = 2
! ( A : A4
. ) A4. G ! A4
6 3 ! !
= !
0 7 A4 : a b "
a3 = 1, b3 = 1, (ab)2 = 1.