% H - 7 , H 1! Z2! E4! D8! P
& H 3, 1, 2
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! Φ(P ) = Ω1(P ) = ap × bp
! " P p+1 H abi i {1, . . . p−
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1} ap × b |
pn−1−i |
× b |
H Ωi(P ) = a |
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(( H 2, 2 |
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3 ( (-./ (. H 2, 2 |
! |
Zpn n ≥ 0! Ep2! Q8 |
! |
Zpn × Zp n ≥ 2! Mpn pn = 8 |
! ! |
D8! Q8! Mpn (n ≥ 3) |
!$ |
Ω1(P ) Z(P ) m > 1 n > 1 |
! |
7 P = Q8 Z4 - H 0 |
! , D8! 0 ! , Z4 × Z2! |
! , " Q8 m(P ) = 2! rs(P ) = 3 |
! ! |
7 P = D8 D8 (/ ! |
, D8 Z4 ( Q8 Z4)! & ! ,
Q8 × Z2D m(P ) = 2! rs(P ) = 4
7 P = D8 Q8 5 ! ,
Q8 Z4 8 ! , D8 × Z2D m(P ) = 3!
rs(P ) = 4 |
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§18 |
" " |
% H I C E ! p = 2 C = Aut(P )! |
p > 2 C = |
α ! α E , pn−1 # |
xα = x1+p |
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§19 |
# " |
g E |
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H -.! 05 |
# 0 H G 0 H & .
.%! (/ 0
(.! 5 & ./
# ! Diag2(q)! U T2(q) × Z(G)
# # . H Z "$# q2 −1 4 q q2 +q+1
4 q
# |
Z p pm + 1 |
# ! |
. H U T3(q) D! D Zq−1 × Zq−1 0 H {t31(a) | a Fq} |
#$ |
CG(H) = Diag3(q)! CG(H) ∩ SL3(q) Zq−1 × Zq−1 |
# |
GLn−1(q) |
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F + |
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+ # (8 &( ( |
§20
$ . .! .! .! ( H
4 E . D D∞ E
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$ l2(G) ≤ 2
$ . H Oπ(G) × Oπ (G)$! . H G = A4 × S3
§21
; D8
2 H Z2 × A5
. H G = S4! B = a 4 b 2 Syl2(G)! A = a ! H = A4
. H G = S4! B = a 4 b 2 Syl2(G)! A = {a, a−1}
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(χ, χ) = χ(1)2 |
χ(1) = 1 |
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; $ S3 ...6 |
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( ,$" ωχ |
..%( ) χ(1)χ(gm)
# χ Irr(G)! ...%
$ . H ; ψ Irr(G)! Irr(χ)ψ Irr(G)
|Irr(χ)ψ| = Irr(χ)
3 " ..&8 ! χ(X) = {χ(g)α | α Aut(Q(ω))}!
" A
, ! ,
! ..00 (% ./
k
! aχ = i=1 χ(gi)! g1, . . . , gk E
4 : G 3 : " #
$ E $ $
"$ Ker (AGT ) = Ker (A)G
§23
+ # 0 (5
+ # .0 ( .0 .
( H E .( .6 .H G "
M = a × Q1! Q1 ≤ Q! 4 M E
G a 3 ! Q1
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! Φ(Q) = 1 > Ω(Q) (Q) Φ(Q) Z(G) !
#
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t := [g, a] q ! ! t Z(G) 2 ga = gt! ga2 = gt2, . . . , g = gap = gtp! tp = 1!
+ # † Wh zQi* .5% mh`*~dh*` .0j .( .- = G E ! G Zp ×S! S E p
G p! E p
I G ! G =a p2 Q! ap Q E ! G = a p Q! |Q| = pq2ba Φ(Q) G/Φ(G) E pqb
" G = Z(G)S! S E
@ pn pq @
G = D18! N Z3
H @ H H 2
# @ p2 pq! 9
a p Q |
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G |
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#! J J # #
#
G D(A)! |A| 4
+ # .( .6! .( 0% .( 0% l2(G) ≤ 2
! + # .( ..! .( .5 .( 0( . l2(G) ≤ 2
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# H G , Z3 S3 H G , Z4! E4! A4
D10
+ # (0 5%! ./- ./ (/
+ # .0 &0
