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升级    52% TA的每日心情 | 开心
2012-1-13 11:05 |
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签到天数: 15 天 [LV.4]偶尔看看III
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对着S4群表看下面就能懂了，我曾把26字母乘群表带身上2月多
& J( B* H9 d' ~, i% L2 L7 V% z
. h* Y# Q' Q8 e5 g+ W/ RS4 := Sym({ "a", "b", "c", "d" });
! X, h* a/ w. G2 y: C> S4;$ Q$ P2 V7 g7 P8 U) s* o, t! q
Generators(S4);3 v$ v7 }8 a- G, F! a
IsAbelian(S4);不是交换群! O$ ~# V4 u, n( U; I
Subgroups(S4: Al := "All") ;列出所有子群
9 e# ^* |/ c4 M0 G0 }/ g Subgroups(S4: Al := "Maximal") ;列出所有极大子群
P& e% [% ~8 M) x% a! I7 ], H7 @2 g4 H' o- x- u# ]+ g* U& C$ g7 o
SubgroupClasses(S4);
& [# Q+ c3 n+ h; u- v& h
4 ?+ `+ W9 L& x1 D U9 L0 O0 JNormalSubgroups(S4);
# C8 m8 U A7 g( W9 F6 l" }AbelianSubgroups(S4) ;
- T2 E. n* f9 X2 \, X# [2 l; H. _5 HMaximalSubgroups(S4) ;
# c8 O3 P m; n7 b, o R2 W# m
1 [1 b, M: a1 Q( ?SubgroupLattice(S4);成格，你可画下这群包扩子群的图
3 Z0 w9 w/ f! ?! ~1 y
+ f% w2 y0 F# W, Y7 `; V+ b" S7 VGSet(S4);
9 X2 d/ X2 k9 a0 l: l; D' BConjugacyClasses(S4);5 [) Z/ _3 _. e( ~
NumberOfClasses(S4) ; 5类% T7 u7 H5 f4 M1 o9 P$ K# Z
8 M: ~+ I9 [1 y O9 ^; N6 d
Symmetric group S4 acting on a set of cardinality 4
# k; j7 d# O6 C" q7 { @ BOrder = 24 = 2^3 * 3/ g# w6 G- q, h. }1 L1 y
{
$ j. A# ~0 E, [: ^- e4 K p+ w1 J (c, b, a, d),% I2 c6 r R) S6 `; z
(c, b)( N* V' F: h R# J
} 两生成元
& D( E% O) X) G2 _+ N1 r/ Ufalse
7 O2 z0 a5 w" XConjugacy classes of subgroups 子群共扼类
8 t3 `7 D- K: _( n$ Z------------------------------
. T4 R9 y3 m @, a" i5 B" Q* u/ \! g2 Z( y5 h
[ 1] Order 1 Length 1* F" C9 }* k5 f3 ]5 }7 ]- b/ W
Permutation group acting on a set of cardinality 4/ @% v* D% j9 p5 I* F0 @! O
Order = 1! e" k; p0 R, u1 R; c7 G; k
[ 2] Order 2 Length 3) R B# L2 z) B2 E
Permutation group acting on a set of cardinality 4$ V& s0 H4 m5 c/ C! I& ?6 S4 W5 a
Order = 25 M: V; {: }! `0 h$ \% K
(c, d)(b, a)
2 v3 R* N$ Y7 E( w9 F$ a[ 3] Order 2 Length 62 w6 f W2 Z6 z; y. u
Permutation group acting on a set of cardinality 4! C1 j/ o" |0 |1 a/ f& q. ~$ }
Order = 2
3 E% b" V" f. B' U# \ (a, d)' ?* Q* r5 j, a5 d, P( ?7 T- ~$ V, |
[ 4] Order 3 Length 47 v( {, b% h) l. n/ M$ v. c
Permutation group acting on a set of cardinality 4# B+ D' _: Q% ~6 X
Order = 36 g6 z! c+ ?( b7 b7 |2 a4 ]' Z
(b, a, d)
; ] w6 L7 y% X6 w5 E( Y[ 5] Order 4 Length 1: _4 G* Z+ Y; v
Permutation group acting on a set of cardinality 4$ X8 {* z/ N4 ~6 i, ?9 u/ e
Order = 4 = 2^28 H1 r# S1 v# G- n$ l
(c, d)(b, a)) I( M: b X4 c7 V2 [- P
(c, a)(b, d)
2 G4 ~/ V, D3 t* U[ 6] Order 4 Length 3& I4 E2 l. O+ D4 \8 R1 C! n
Permutation group acting on a set of cardinality 43 G# s8 R" {; ^( T6 b
Order = 4 = 2^26 r- m# p0 V! A" z3 E6 _; @/ G
(c, d, b, a)
5 q8 s) g7 X! L! i* V. L+ w (c, b)(a, d)1 {9 p4 Z0 d- T
[ 7] Order 4 Length 3
8 ? ^1 u# q6 ] Permutation group acting on a set of cardinality 4
8 u6 ] `% W0 \: `- h Order = 4 = 2^2
& K3 m9 [! z! F7 ]5 F" S2 r6 _ (a, d)
+ A( I' ~7 t0 I8 }" Z (c, b)(a, d) ?0 T6 L2 V! S+ T! N' Y
[ 8] Order 6 Length 4
, J7 h4 i8 {4 ?' s8 G0 D Permutation group acting on a set of cardinality 4. m, X8 J5 a$ T4 V
Order = 6 = 2 * 3* h) i+ u+ l+ H- B
(a, d)
! T8 N* ^$ z0 D% E9 v6 f: P (b, a, d)1 O7 Q, V8 o2 W, B6 e
[ 9] Order 8 Length 3
- o" B2 M. S$ O, z! H! a Permutation group acting on a set of cardinality 4" A& V9 l1 ^; K% d0 \
Order = 8 = 2^32 E( E0 R; L4 b8 x
(a, d)# ?7 o8 Z4 O$ I
(c, d)(b, a)+ u8 r4 b/ N! a2 a: s% m
(c, a)(b, d)) H5 w6 S+ S* E s: }1 ]' u
[10] Order 12 Length 1
; n/ @7 h: k. i. R Permutation group acting on a set of cardinality 4
, a) |7 t) i3 n6 x) v* Q+ _ Order = 12 = 2^2 * 3
* P; u% w) g# S* F$ \; O% r" i (b, a, d)( H3 R9 m/ n' l. H
(c, d)(b, a)
: U! b% z3 n' o# c. h6 ?9 i (c, a)(b, d)/ Q. `6 v& j6 {! f
[11] Order 24 Length 1
2 _0 w: U; J0 p2 g" Z7 L Permutation group acting on a set of cardinality 4
# S/ m* k+ y. a- J. i# f* t d c% c Order = 24 = 2^3 * 3
. i+ m! D; y. x/ A' Y0 F (a, d)" Q& s0 \9 S5 A8 `7 D8 p( J
(b, a, d)
% k0 S# d* _8 k3 P (c, d)(b, a)! q: t$ k) [' z& o: e
(c, a)(b, d)
+ Q8 ]0 {2 f9 ^0 N( ]% W5 SConjugacy classes of subgroups
* t: d, e$ n& u( F3 Z6 F------------------------------
1 S( Y1 f9 i3 j m" ]! V C( b) a3 c1 l
[1] Order 6 Length 40 E2 x7 _4 x# ]- ~6 F% x
Permutation group acting on a set of cardinality 4# m* o9 s/ `/ y' s8 x4 |# X
Order = 6 = 2 * 3# K. ]. k5 G$ m! L9 C8 P
(a, d)
+ k8 v: o( e! S6 {9 D; x! t3 b! a (b, a, d)# w9 s5 K- z+ T2 W
[2] Order 8 Length 32 n5 U) j/ @$ q8 M
Permutation group acting on a set of cardinality 40 L5 V0 T6 P7 j- M$ b, i
Order = 8 = 2^3
1 }, e; P( N. Q7 z1 U1 J0 N (a, d); a' ~* H# Z9 Y2 l3 d6 @6 y( y
(c, d)(b, a)
3 C6 E) f+ l8 P) {2 O (c, a)(b, d)
' W- P0 z; ?4 w& C4 C, r ?[3] Order 12 Length 1
' E5 E- ]7 E$ n8 D Permutation group acting on a set of cardinality 4; R' B" n& n4 t% h
Order = 12 = 2^2 * 3
$ t/ _. E9 U/ F (b, a, d)% t- s% p. ~* S7 B# h
(c, d)(b, a)8 T! s9 e( M' s6 T8 Z
(c, a)(b, d)' t$ I- l/ L" u0 @# z1 v
Conjugacy classes of subgroups: R+ z+ q+ t+ g* \7 ]
------------------------------
% \1 Z/ t! g: g+ ^, S: A7 i) v+ Q
[ 1] Order 1 Length 1" Z4 @5 o- \# F: _( \/ C% Y
Permutation group acting on a set of cardinality 4+ r" F- H2 n3 y8 f n+ w4 u
Order = 1( K- b( ~6 P5 J( t2 B
[ 2] Order 2 Length 36 |2 t! j5 y/ ^- Q( y& s) _6 l5 z
Permutation group acting on a set of cardinality 43 E. v& Q* a" }. R
Order = 2
$ n2 g' \+ f' b' ?& k (c, d)(b, a)
, k' L8 b4 ], k' ]- b; \% r[ 3] Order 2 Length 67 q" a- t _# X0 }
Permutation group acting on a set of cardinality 47 K$ ]/ S; s7 Z( _/ T
Order = 2
, m; n" J t+ Y) x+ T n (a, d)! o) D+ G9 h3 G$ V* Y5 d
[ 4] Order 3 Length 4 S- ? v; Y8 ] ^5 ~
Permutation group acting on a set of cardinality 4
( n6 y9 T) g M1 }2 Q Order = 3
) W! L5 o2 a) @9 Y; o* ]3 F; D# D9 x3 G (b, a, d)8 b) a" P8 W j! [
[ 5] Order 4 Length 17 G, t2 S& E' {' [- r2 A
Permutation group acting on a set of cardinality 4
& \+ @3 V* t) X& S. v- s Order = 4 = 2^2
; t# N: {% ^0 J (c, d)(b, a): R( N( `4 r9 d( c l3 w- \" t( `
(c, a)(b, d)7 x( j' j7 Q3 @1 A% k3 ?
[ 6] Order 4 Length 3
4 a# U5 K. F9 H/ u Permutation group acting on a set of cardinality 4
1 z9 S* |) |6 D: a# [ Order = 4 = 2^2
5 r/ z, t& v! q, k0 v/ h, m( ?8 ~ (c, d, b, a)
# b2 l- y0 N3 c (c, b)(a, d)$ \/ P$ P5 }+ u3 r; ~, U5 v' B
[ 7] Order 4 Length 33 g& D; _3 k/ r( r1 v; s
Permutation group acting on a set of cardinality 45 s2 a$ F$ j( ^3 x% _4 X! y& C* z
Order = 4 = 2^22 o& e# D Q8 a8 f
(a, d)1 S+ z* z: x ~* F
(c, b)(a, d)$ G, }: f* q) H
[ 8] Order 6 Length 4$ [5 H, l; g8 c3 Y8 Y
Permutation group acting on a set of cardinality 4
: z9 A0 _; z2 w" n Order = 6 = 2 * 36 K0 a' q/ l v2 z* c! Z
(a, d)1 U1 Z) S7 f# b7 @% @
(b, a, d)
7 u* j- ~% T- G3 r, Z$ `[ 9] Order 8 Length 3; _; l! a7 Z7 Q% n" J l) S
Permutation group acting on a set of cardinality 4
3 s9 w, z: j- f Order = 8 = 2^32 T* ]& l' I4 a m) V3 W0 c$ e( p
(a, d)- @: i# x% i: s: r
(c, d)(b, a)
0 ?5 C: y6 a5 A& t (c, a)(b, d); \5 ?/ i: t+ r4 h6 [
[10] Order 12 Length 1! U+ e6 ~: a: {* L( p0 o
Permutation group acting on a set of cardinality 4: z( L, y4 e& m4 d: `
Order = 12 = 2^2 * 30 @! _1 y2 B7 Y3 j' p
(b, a, d)4 E! w: \ Y! a
(c, d)(b, a)
8 V0 C: O7 m7 w' M+ K (c, a)(b, d)
# n6 @) W- l$ c! c0 @[11] Order 24 Length 1* c. e8 T1 j4 O+ \0 L
Permutation group acting on a set of cardinality 4$ a1 \5 r- J4 p6 N
Order = 24 = 2^3 * 3
9 v3 |/ o# S b- W! A (a, d)
1 U$ p4 f; Y5 _$ J8 J (b, a, d)
! ~" c% T6 `; F& j+ C0 v/ w (c, d)(b, a). X% ^# c) |/ l+ E1 M$ c; q
(c, a)(b, d)
( {+ r! G. y7 _) L( G" LConjugacy classes of subgroups
' ?0 B0 q m p$ G( _2 U+ ^6 v% C------------------------------9 ]5 i5 y& B8 V( ]# \
* H( L. r3 ~9 \% V; U0 Z[1] Order 1 Length 17 b( v+ n, D* p9 t
Permutation group acting on a set of cardinality 4
* a8 J# Z2 K. b7 s; P5 i/ k Order = 1: R3 Q1 n3 w( M B3 s5 s
[2] Order 4 Length 1' r- z" j( b a- E
Permutation group acting on a set of cardinality 4
3 {. u1 w. T! @5 ^6 T- R/ }. ` Order = 4 = 2^20 l" ~ ]" @6 B. q$ i& w
(c, d)(b, a)
+ Z5 l/ d6 f- U1 Y1 D (c, a)(b, d)
7 [7 n7 C- a: @) `# s; p[3] Order 12 Length 1+ p! i* `; R* k4 x. C
Permutation group acting on a set of cardinality 4 B- v9 ~* `+ D; J9 N6 d
Order = 12 = 2^2 * 3
b0 I: ^8 x' C* ] (b, a, d)
' N. D9 _' ^- r; g1 [6 R; x5 p# n (c, d)(b, a)) Q2 u$ D0 k" B8 h- b: J0 T* `
(c, a)(b, d)5 l# S! p% y$ m6 \( }# v$ |
[4] Order 24 Length 1; a @( H% e: }$ k$ [4 _9 U/ `; q
Permutation group acting on a set of cardinality 4
! N1 R. H8 L h+ g' p2 |; p2 O Order = 24 = 2^3 * 3' z7 ^6 w2 L, _& l* A1 H
(a, d)
' u9 z& s9 V4 l6 P, M (b, a, d)% K7 f! e! j* O& w
(c, d)(b, a)
8 ~# \) @ _& b' ~/ O& s3 K (c, a)(b, d)
; o. M; a: f% X' |# PConjugacy classes of subgroups
. q% g2 n3 l2 `: o( g% o------------------------------! S0 C( x/ b7 k
z, Z3 [+ E4 ]4 E
[1] Order 1 Length 1# q/ v' k) i4 S7 P6 ]. }
Permutation group acting on a set of cardinality 4
( a" y: B7 n. Z Order = 1 ^& q* S- V' G) t
[2] Order 2 Length 35 i5 P% \$ P7 v) w0 U d: R$ t
Permutation group acting on a set of cardinality 4+ }+ t& M! G- G5 q
Order = 2
' i7 k: m: d0 L (c, d)(b, a)
( q, o( i4 g% S! M* y0 O[3] Order 2 Length 6
8 ]" C+ `7 Y7 b Permutation group acting on a set of cardinality 4) I8 W; q' V' G- `. I
Order = 2" h7 k5 x1 Q, \+ C/ J0 r; a
(a, d)- F" n) _2 }) d. G2 ~' W7 v
[4] Order 3 Length 4
. H& `$ z6 E$ q Permutation group acting on a set of cardinality 4: u' q% {0 `2 z: \/ v8 o6 k, ?
Order = 32 R4 P9 H+ N& ?* p9 r* }% {
(b, a, d)5 E7 X3 W9 J6 h! j. L9 q: u5 E
[5] Order 4 Length 1
/ L. u; v# X% R2 Y4 Q, Z! D( ~ Permutation group acting on a set of cardinality 4
6 \- y5 b& M3 w, [& f Order = 4 = 2^21 n0 d Z0 k( Y4 D
(c, d)(b, a); S( V o- R8 U& T0 S" q5 S
(c, a)(b, d)
* F- _ p+ \# \" J8 \! H+ u[6] Order 4 Length 3
, M, Z$ y$ R. P$ ^8 F; C Permutation group acting on a set of cardinality 40 ~; G; F$ L5 i: g
Order = 4 = 2^21 i1 A5 B0 {' a! y+ k
(c, d, b, a)
: m; b8 u/ a" j3 I$ h4 s (c, b)(a, d)! ~" d: x7 q% \8 `
[7] Order 4 Length 3
; j' v# S* C& @" v. k Permutation group acting on a set of cardinality 4
6 E2 a- J+ f0 \7 D Order = 4 = 2^29 E) t. a3 f4 W# S: T* ?
(a, d)9 Z' z, |& b3 S3 h/ t
(c, b)(a, d)
6 }5 P# C4 ?. _5 z; a8 y! a5 nConjugacy classes of subgroups8 M* f& w2 U0 n3 i
------------------------------" f8 Y9 q2 S$ f, _6 ?0 S, X
# p0 e) @( o; A[1] Order 6 Length 4) t$ F3 [( ^, v ?& v0 a
Permutation group acting on a set of cardinality 4' L, T$ G# M, e+ F. D) |* {
Order = 6 = 2 * 3
7 r6 }& O" u3 f8 z! f (a, d)6 c+ b- M+ A; Y t
(b, a, d)
4 ^. {$ b5 f6 F% v[2] Order 8 Length 3
x4 G# B! A+ L8 ~4 D5 G/ k Permutation group acting on a set of cardinality 42 J: T" n4 s; y" m! m+ W2 p' X
Order = 8 = 2^3
3 I. o! M/ ^9 y0 s$ C) n (a, d)
( Y! a5 u y: Q+ X! o+ V( [ (c, d)(b, a); y! ]% V5 L% h
(c, a)(b, d); k+ L: V/ @3 z N7 D
[3] Order 12 Length 1
' l: E9 A3 q' a, }# J1 L% j/ ^ Permutation group acting on a set of cardinality 4
8 L+ W3 B: H1 k$ g- t Order = 12 = 2^2 * 3* k: `# h& b; T* U& T
(b, a, d)
# w8 Y7 Q. F+ M& u/ U: U (c, d)(b, a): J: q4 \/ i" V
(c, a)(b, d) p8 _$ q8 k, O$ u* n& Y
5 z; F: w4 w* g7 T
Partially ordered set of subgroup classes7 H- r/ u/ E; ?3 Q# ^
-----------------------------------------
3 @) ]$ }, E4 g5 D4 x# {
- k! N1 |: F0 [* n' m[11] Order 24 Length 1 Maximal Subgroups: 8 9 101 a. F# J9 M$ G% E- E# }& u: x
---
% R2 I' G! {) I7 I, O2 C[10] Order 12 Length 1 Maximal Subgroups: 4 5
2 z; _: T/ [7 q) n7 A[ 9] Order 8 Length 3 Maximal Subgroups: 5 6 7
1 N" U& `) l; A+ d& b---
+ T5 J) X, \% S6 a" l. W: J q[ 8] Order 6 Length 4 Maximal Subgroups: 3 4
, ` `, Z, E. d[ 7] Order 4 Length 3 Maximal Subgroups: 2* i1 ^( ?# w; {! O- ^6 S* d* |
[ 6] Order 4 Length 3 Maximal Subgroups: 2 3$ ]- [- V# e; h3 P
[ 5] Order 4 Length 1 Maximal Subgroups: 2. c! }- R9 K) b c1 l
---; a0 d; U$ V" s
[ 4] Order 3 Length 4 Maximal Subgroups: 1
# a" u0 w" V) H& e3 {! \[ 3] Order 2 Length 6 Maximal Subgroups: 16 _4 z7 Z( }% {1 ~
[ 2] Order 2 Length 3 Maximal Subgroups: 17 z9 S7 o# V8 d1 N* @
---
9 z# u8 v3 Y; t- ^3 w[ 1] Order 1 Length 1 Maximal Subgroups:$ X9 r- m- L8 ] u; U7 P
, H; L2 t: V/ F6 s8 i9 D$ ?6 G
GSet{@ c, b, a, d @}# U1 ~ s0 g8 z# u
Conjugacy Classes of group S4
: r0 b9 w9 R( u6 \" }-----------------------------% X/ m' Q1 F x" u% v9 t" X# {+ c
[1] Order 1 Length 1
% Y: Q0 @: _, X* I Rep Id(S4)
" g4 `( @1 d% p, L$ }% M, m, H
; M6 [/ o1 F: O[2] Order 2 Length 3 {5 f, L5 o( b( ]$ h' f8 _" v
Rep (c, b)(a, d)
" [$ T5 F9 t" j
2 S' N( U0 J9 f6 D Y[3] Order 2 Length 6 6 `! @4 n8 ?) }3 ~6 ~. | J
Rep (c, b)
3 U5 l/ f5 |* v& I" E
% k. [) {0 _8 w2 h( S- g1 g[4] Order 3 Length 8
9 K9 {2 f7 R% B6 {$ C+ r0 K/ z Rep (c, b, a)
' `# Y$ J, d3 z& q7 `
$ V; Y o C1 m[5] Order 4 Length 6 1 ?4 { h( w" j! U9 F9 u* G
Rep (c, b, a, d)
6 F4 h7 F" f5 u# q1 P, D/ Z L' a: ?
4 h4 d- O# n& a+ e: |8 M$ W. E5 |
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发表于 2012-1-1 12:28
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