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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- B9 _( A2 y- l9 g5 h: ?
( C% ?- x; E: |5 _+ D5 C+ oS4 := Sym({ "a", "b", "c", "d" });; h" ]; M8 x+ K6 y+ M
> S4;
1 {7 ^/ b. q* z2 Y, ZGenerators(S4);( z7 p7 U. x' b* t) ~
IsAbelian(S4);不是交换群; g+ ]# F2 H; R2 x
Subgroups(S4: Al := "All") ;列出所有子群
( N) T! y1 h1 d. B4 t. K) e Subgroups(S4: Al := "Maximal") ;列出所有极大子群
' U! z9 d& K) E8 [* t- f3 V5 T- S
SubgroupClasses(S4);
2 C9 e3 U; u5 [; S
. P- E& i" N* G) b! W+ N5 K) D& CNormalSubgroups(S4);
% U# K m8 Z1 Z- j3 X+ EAbelianSubgroups(S4) ;/ Y2 f$ W( x" Z& w
MaximalSubgroups(S4) ;
% t3 Y5 ^# w; G1 P6 V3 b& g$ g4 q- B N" h
SubgroupLattice(S4);成格，你可画下这群包扩子群的图8 G! O- }# ]& }8 I
$ q7 W; p) ^$ q4 _GSet(S4);" i3 t7 ] y% ^' m% z0 o( U
ConjugacyClasses(S4);
( \ j# _4 U6 l6 j9 x8 tNumberOfClasses(S4) ; 5类
3 t0 }9 V- I# @% n. I
5 ], M5 [* f6 l4 e& _" e( `7 `Symmetric group S4 acting on a set of cardinality 4
Y7 q9 N+ \( J- y7 P, rOrder = 24 = 2^3 * 3" R! X( m* k- V2 A
{0 q6 E3 {0 e# p/ P( D
(c, b, a, d),; r! v# H$ d ^! h( G7 K- D$ r: o
(c, b)
/ a. A3 X; ?7 I- ~: P* c" T! ?! y- |} 两生成元
' n! o/ |) i+ efalse
. ?) _, N4 T" Z" Q! a. i- AConjugacy classes of subgroups 子群共扼类6 E8 Q6 b! i) R5 K/ \8 K& O
------------------------------8 G3 k: Y& E8 i6 u( a, q
' [; q- V% q0 g' Y" w y4 l[ 1] Order 1 Length 1
$ h4 Q3 v7 t- A6 u3 H# \# f6 u Permutation group acting on a set of cardinality 4& W. q0 Q, m) [- A) f% z
Order = 1- X5 @* |/ R' N% `+ }) J$ F% @
[ 2] Order 2 Length 3" Q5 X5 F- _0 O% E
Permutation group acting on a set of cardinality 4; ^; h+ S2 ^- G, e- l: }" [
Order = 2
3 G% O7 t, K$ \& I (c, d)(b, a)0 \1 }4 K6 E9 W" S% x/ e0 S" e
[ 3] Order 2 Length 6* m; f4 v) K0 E! g2 j4 J
Permutation group acting on a set of cardinality 4
6 z( z5 u3 V6 g" E Order = 2
: }9 }/ |) p, i F( }! g' _ (a, d)6 Z6 U, |( C; k& B
[ 4] Order 3 Length 4 d4 z2 Z& P% j5 o$ j0 a
Permutation group acting on a set of cardinality 4
$ k8 h5 ?1 S2 O8 N Order = 3: j, S1 P1 u a$ M( k
(b, a, d)
; ~# f+ l- N" L; w m. C5 f[ 5] Order 4 Length 1
2 X2 K2 d( k: b+ [1 d9 `* M8 f, g Permutation group acting on a set of cardinality 4* J1 K5 A/ V% O) q3 I9 \
Order = 4 = 2^2
# x. ^. E& O2 G' z (c, d)(b, a)5 ~/ D. I/ r J# U: j- `" a) r
(c, a)(b, d)
7 k \+ L: M( |- M[ 6] Order 4 Length 3
7 T, [% g- m& o \ Permutation group acting on a set of cardinality 4& \* w; t4 P8 r2 z
Order = 4 = 2^2
! `' E7 N7 ?: A( C+ s (c, d, b, a)+ _3 d- F( g$ y; m; p
(c, b)(a, d)' f, X2 D4 K: m! B0 i5 B0 e- H6 x
[ 7] Order 4 Length 3, }" Y1 g, Q; L( z6 q
Permutation group acting on a set of cardinality 4
) V6 |/ p) p" i$ y0 U) J Order = 4 = 2^2" r4 J k1 V& ?$ I
(a, d)
1 l8 y. P5 z5 ?# q (c, b)(a, d). r# q& o% x1 z+ J; s
[ 8] Order 6 Length 4% W2 }' K8 |4 O
Permutation group acting on a set of cardinality 4
: e7 ?' h4 n/ R9 _6 d9 S( T Order = 6 = 2 * 3; g7 Y- P0 a7 _+ p
(a, d)
3 V0 M1 p: A, l (b, a, d)1 z7 F' S8 X* [: H% @
[ 9] Order 8 Length 3) F4 _2 l: A( v
Permutation group acting on a set of cardinality 4. e: c7 Q! B. z0 L: \+ y: V/ M2 j P
Order = 8 = 2^3
9 @( C3 `# H+ Q. [5 Z (a, d)
- ?5 N- y( o, `" A. s) M (c, d)(b, a): V( f* h+ c* {. {+ D6 }4 w
(c, a)(b, d)
2 `2 p9 \0 G2 A4 t; F! v L[10] Order 12 Length 1
( E4 ^* c9 i/ |1 |! x Permutation group acting on a set of cardinality 4' E' Y0 x, X2 d/ A) q
Order = 12 = 2^2 * 3% ~6 s z0 W* k6 y: `9 y
(b, a, d)
2 ~* D1 k* H6 B7 A9 f3 M& O (c, d)(b, a)
$ p( e' D1 P& b# v (c, a)(b, d)
+ W+ l( E8 Z9 p$ ?4 d9 q0 t% y6 f[11] Order 24 Length 1
`, E, F t* T0 I) ~. Z Permutation group acting on a set of cardinality 4' b) O9 n; J$ x! r. ^
Order = 24 = 2^3 * 3 ]/ l% c1 q! m) A; R ^
(a, d); z" ~7 {9 Q5 h- m
(b, a, d)
8 e* d9 [+ ?. s" s* l! |2 b, V4 V (c, d)(b, a)2 f% J3 |9 G" R+ L; |5 s. F; w& _- l
(c, a)(b, d)
% p$ |) T% }& [: ?. u2 S0 K6 K. fConjugacy classes of subgroups& N; \; T3 ^; F$ K2 `1 e5 I: _
------------------------------- b" n6 G7 F# W- a+ |
0 M) C& L1 k% d. f. f9 }
[1] Order 6 Length 4
8 s2 @: F% h J7 n- h- Q6 A Permutation group acting on a set of cardinality 4# t5 \1 R" s R
Order = 6 = 2 * 3
) Q6 d% Y6 G2 N. G, e' Q% C (a, d)
, q7 x, m% _) c( W (b, a, d): y; j0 k; S$ @+ i
[2] Order 8 Length 3
% P" L8 Y( S1 t Permutation group acting on a set of cardinality 4; S4 H) X% A/ Q" m1 |4 g$ Q
Order = 8 = 2^3; z) t' q8 P* d4 V
(a, d)) r' s( y: L' h$ L/ Z
(c, d)(b, a)/ j; q( p O7 ^8 c/ q. c
(c, a)(b, d)
( w+ N7 B8 S: `* L* ^[3] Order 12 Length 1# u2 {. c" ~4 A5 X
Permutation group acting on a set of cardinality 4
0 i$ a7 a( t/ S" F' W Order = 12 = 2^2 * 3
3 E) n1 c2 }' `$ ] (b, a, d)
' ^- M) J$ P3 H4 j$ ^- s (c, d)(b, a)
' T! Y# x" f, t; g4 O0 {/ w9 ~8 ] (c, a)(b, d)
[4 g. n9 n# ?$ |! |8 ]Conjugacy classes of subgroups5 K: W6 B& P+ e* @7 N
------------------------------3 o: e4 R: \ u, n5 d. k/ H( ]* s
6 E- e$ X5 o: j" q, ~, y[ 1] Order 1 Length 1% f) a9 K/ p& r% w5 w% P L
Permutation group acting on a set of cardinality 43 g* K3 C; C# z5 y
Order = 1# l( S2 h. f3 S3 B, z
[ 2] Order 2 Length 3
0 H3 o0 w. I8 j Permutation group acting on a set of cardinality 4
4 O- B6 d/ i5 M V" b, I# @ Order = 2
: R3 u+ P& N2 j (c, d)(b, a)
4 K+ V+ _: e2 x3 |[ 3] Order 2 Length 6+ o1 P' A; U1 B$ W' Y/ Y& s! t, D& O
Permutation group acting on a set of cardinality 40 I" `" H, x9 s3 t
Order = 2
( O/ ~ B" z/ u (a, d)
' t/ g% d' ]1 R, a. C[ 4] Order 3 Length 4
+ W7 W% V2 h% r4 L/ _ Permutation group acting on a set of cardinality 4
; k: c; X( \) A6 @0 T' @) q Order = 30 M2 B; ]$ p8 e2 U) G" ~
(b, a, d)9 i+ e0 l4 O0 F7 c
[ 5] Order 4 Length 1
8 T2 W1 N1 g, a* ? Permutation group acting on a set of cardinality 44 O# w+ \7 i' u' j7 s, t E/ }& _
Order = 4 = 2^2
5 m8 x+ b4 J- Z6 z1 V! ] (c, d)(b, a)' {5 U$ c6 l' o# R$ j
(c, a)(b, d)
8 R- u W6 v s1 z7 r8 N0 L+ C[ 6] Order 4 Length 3/ c( p' n& y# W
Permutation group acting on a set of cardinality 46 N3 ^: [8 k6 F* }, s
Order = 4 = 2^2
- K# g! q' L4 Z. N- k (c, d, b, a)
' \. P$ n1 U- p (c, b)(a, d)
4 Q+ S+ A: _9 ?* z# j[ 7] Order 4 Length 3
/ ]5 c5 K- O# W( H% O) l. N4 S2 | Permutation group acting on a set of cardinality 4& y/ D% R0 T: a7 N$ F
Order = 4 = 2^2
$ H: N/ V9 V# W+ m* \9 o/ ] (a, d)6 m6 S* C e( X* y& R2 u8 p
(c, b)(a, d)/ ^! w. I9 c6 n
[ 8] Order 6 Length 4
1 f% A2 s1 W/ z3 g+ U: Z Permutation group acting on a set of cardinality 4
& d* c5 N; J0 j/ y& u' p O Order = 6 = 2 * 3/ U1 Q" j$ c5 M7 t9 b' h
(a, d)
. }" |2 k7 v$ \8 u) i& ? (b, a, d)9 M6 S! N# a* u# F& ^
[ 9] Order 8 Length 3
3 W( v/ a& g. N4 n: l/ C0 W; m6 z l Permutation group acting on a set of cardinality 4
* v( x+ s# q/ A Order = 8 = 2^3
1 y: n2 y& f3 Z (a, d)8 G; c* C& S2 Q: K
(c, d)(b, a)
0 L, ]# n# Q. R4 L# A2 X3 P (c, a)(b, d)
$ @1 K' F- R( |[10] Order 12 Length 1
0 E* O7 B0 z2 ]4 v Permutation group acting on a set of cardinality 4
5 l C6 O7 D( p Order = 12 = 2^2 * 3) g" Y U/ P4 {9 l/ K) X7 ?
(b, a, d)
9 l. [& v8 u$ K (c, d)(b, a)
" P5 Y/ R* D7 E/ R. l9 I* n (c, a)(b, d)# |6 S4 V/ @# ?5 Z* c6 Z8 N
[11] Order 24 Length 1" q1 m6 b' W+ X: t3 ~
Permutation group acting on a set of cardinality 45 x" G& ^, h' y" f, K4 V% S
Order = 24 = 2^3 * 3/ F3 I4 p$ k# z" T+ m) Q
(a, d)9 s4 ^$ M* z8 G6 c
(b, a, d)% O" E' g4 a7 v( I2 [3 S* L
(c, d)(b, a)
7 x6 X* s& f/ K (c, a)(b, d)3 s4 @2 R7 r5 F: ~/ \5 u8 u/ e
Conjugacy classes of subgroups% m$ e* X/ }8 k
------------------------------
5 @. V4 b4 ~& i! z
4 _# e' T$ [1 A3 A[1] Order 1 Length 1: P5 a3 c. h$ L2 j& T7 D( J
Permutation group acting on a set of cardinality 43 F2 ]7 Q8 q, k: j" v1 Y( L
Order = 1
; d9 ?9 ~6 o( z0 t+ G4 P# ?5 z7 L& E[2] Order 4 Length 1
5 S7 Y. |! G1 _! E! A! ^/ g Permutation group acting on a set of cardinality 4
9 Y4 {: e9 J& T; A+ T Order = 4 = 2^2' P( n/ H! t2 b0 K0 w9 w
(c, d)(b, a): p( C0 W5 `: v9 f g, }
(c, a)(b, d)
- P5 e2 Y# Q; d8 ^1 Z+ \3 N2 A[3] Order 12 Length 14 ]- _: u$ w( \+ ?" a
Permutation group acting on a set of cardinality 47 a9 \$ _7 Z- ?: ^
Order = 12 = 2^2 * 3
( K# o, f7 Q3 P3 I! L# e (b, a, d)
, K( x M/ h. P. b# B2 Y (c, d)(b, a)
# o( H+ h& O. n, p' y2 X (c, a)(b, d)
8 S; n3 j2 \' g; k$ @[4] Order 24 Length 1
7 x) T+ w" H' l; B6 X, s0 e Permutation group acting on a set of cardinality 4
; J" O( }) `. Y8 a7 S4 u* H Order = 24 = 2^3 * 3
: b* f4 y6 m9 Y1 C7 ~ (a, d)1 N. w& L; s6 i
(b, a, d)4 k: i5 g$ J% S2 t6 P2 a
(c, d)(b, a)* k" R, Q* v( e1 R+ v ?# W
(c, a)(b, d)
. L% O% d$ B' G8 a4 iConjugacy classes of subgroups% J3 L1 Y. N6 f1 d# D; Y
------------------------------
4 s0 V( O- v& u
5 c& T, ]" a9 n[1] Order 1 Length 1( S* @/ d6 l) V9 K* r' m$ d" s! i
Permutation group acting on a set of cardinality 4# o4 e& `4 H0 X) H
Order = 12 r5 f: w$ ?, F; c; n2 c Z; `6 v
[2] Order 2 Length 3
. x! x D" R' E! P% @5 r Permutation group acting on a set of cardinality 4
5 |6 k5 Q% D5 D0 X Order = 2
) } }* R' ^: E$ T* ^: Q (c, d)(b, a)' Q: ]* N( B; T
[3] Order 2 Length 62 I& L6 v$ O8 W. ]/ q- s, C) L
Permutation group acting on a set of cardinality 4. Q/ g; v2 k, \. w4 t0 o
Order = 2/ b# s8 R5 O6 c, f
(a, d)
2 m V# v' s2 j+ C/ }" F& _[4] Order 3 Length 4
3 T3 ?0 g+ O* F; X Permutation group acting on a set of cardinality 4' w3 H! V$ a* E
Order = 3
6 F1 y% B/ p$ O4 \: a8 p) A% | (b, a, d)8 B0 S$ ~7 k8 q- f
[5] Order 4 Length 1
; Q; W [" X5 Z Permutation group acting on a set of cardinality 4# {, m. ]. a/ Z# j% L" J9 R
Order = 4 = 2^2
1 |( q* I" g6 N8 W3 h" G3 @ (c, d)(b, a)
1 r6 c |' V& r3 k4 _- |9 ~ (c, a)(b, d), ]1 F& B) ]1 w, `
[6] Order 4 Length 3& W1 d+ r+ ~0 f( P9 |. b
Permutation group acting on a set of cardinality 4
$ {' t6 u D/ \7 D3 Q9 ?, u Order = 4 = 2^2
$ u4 P5 l4 I, \, m2 h (c, d, b, a)
3 R5 s# l2 `' w- F& q; x& ^ (c, b)(a, d)
; A7 }& r) y' J2 U+ E/ @[7] Order 4 Length 30 I, g5 r8 G: U& A/ X
Permutation group acting on a set of cardinality 4
% y& r' P- N2 i! P( q+ `3 w# S Order = 4 = 2^29 A, B1 y6 C, U( a4 h8 s
(a, d)
" Z, x8 C( k; d) h (c, b)(a, d)
9 _4 x; |0 U, P& s( G0 v# Q+ ~Conjugacy classes of subgroups: `' `* X! J9 m7 m
------------------------------* k5 m' B8 y5 P8 Y3 K
: E1 d8 i* ]$ b \) I1 `/ t
[1] Order 6 Length 48 A- X2 C9 _0 }. A' g
Permutation group acting on a set of cardinality 4
& Z1 A; s& n1 l8 C, M Order = 6 = 2 * 3
& A9 e8 k: Y4 r1 v6 A3 j3 B3 J# C/ V& q (a, d)+ l7 X$ l% K, |5 _$ F+ y8 k' F* d
(b, a, d)
0 |- P! b! V% m[2] Order 8 Length 38 z- Y; T2 I3 B# ?8 ^' W
Permutation group acting on a set of cardinality 4
4 i }. K( K! u* F$ j Order = 8 = 2^3
$ l, N6 Z# N, C (a, d)1 V% `5 |$ H _3 O
(c, d)(b, a)
6 R0 d7 C9 M* g. `! f (c, a)(b, d)
. @5 i [: }) c! E6 O- J[3] Order 12 Length 15 ?+ A$ m7 i! `4 n& i6 H1 ]
Permutation group acting on a set of cardinality 4) Q! U/ S1 O6 X2 a& \+ i& h8 z
Order = 12 = 2^2 * 3
6 ^2 r& a; v( T (b, a, d)
1 S' l. `2 d* p, M (c, d)(b, a)/ [7 X' C% n! @ y
(c, a)(b, d)
+ Z9 h7 p. U" R# G; `2 ^4 H, E( B7 d6 _" U2 E0 `1 x
Partially ordered set of subgroup classes
! Z! g3 [1 ` s' Y/ ?: n( e2 z-----------------------------------------
) Q1 l; g" d& Y# E- F7 M6 m; S' r3 o' s/ [
[11] Order 24 Length 1 Maximal Subgroups: 8 9 10
) x2 H. ?6 _9 h5 f d) `---4 j, D K9 @- n0 W9 [, `
[10] Order 12 Length 1 Maximal Subgroups: 4 5
P4 e8 x5 v0 ]8 g2 F( d* q/ ^[ 9] Order 8 Length 3 Maximal Subgroups: 5 6 74 c+ D/ J- h. p
---4 r0 h6 a4 m: Y, \- D
[ 8] Order 6 Length 4 Maximal Subgroups: 3 49 v/ O1 w4 o9 F' G' x" j
[ 7] Order 4 Length 3 Maximal Subgroups: 2
! m5 O4 ^" z4 Y2 z9 v[ 6] Order 4 Length 3 Maximal Subgroups: 2 35 K8 u" V' C7 e3 t$ k3 S+ g d6 S
[ 5] Order 4 Length 1 Maximal Subgroups: 2
: T* [. U6 K. O9 S; D i5 u" c---- Z- U$ W/ S2 D" v0 I" M
[ 4] Order 3 Length 4 Maximal Subgroups: 1
o$ g4 i c" V' E L[ 3] Order 2 Length 6 Maximal Subgroups: 1
! E2 R1 ~3 t$ r: Q$ Q$ u[ 2] Order 2 Length 3 Maximal Subgroups: 1
$ i' Y4 h! f9 `: h7 @# f: R4 y% z---8 c6 W9 f: D5 F' w% R; N
[ 1] Order 1 Length 1 Maximal Subgroups: {4 y- C8 `* ?- Y$ g! F! z+ W
, X* W# p' h) l0 K' ?- VGSet{@ c, b, a, d @}
4 F1 l4 t) V0 f# D6 }Conjugacy Classes of group S4
* _( V1 f2 @. o& L-----------------------------
5 ?; m) g) G/ q) W" e: L2 [[1] Order 1 Length 1 ( Z$ L9 B# b. s0 V- C, K
Rep Id(S4)
+ ?3 y4 S9 @: ~6 m0 ?: M5 t7 v% H! N; b$ l& S7 V1 e
[2] Order 2 Length 3
( Y% T# K( t' ?. |' D Rep (c, b)(a, d), p! _& \* R( ?# S; Q/ r
6 V0 p4 j. E% q. }5 Q0 t# A' n0 s
[3] Order 2 Length 6 8 L0 p0 _( g5 J: R( ~
Rep (c, b)/ r* J$ n$ x* ]3 w! _
: F) G1 s$ x z! Q$ s[4] Order 3 Length 8
* O7 X: w3 J, ]5 V Rep (c, b, a)9 X7 M4 k$ l; ]1 L4 e
3 [! Z! L, w' ~! \) V( g: I, B$ Y# p% K
[5] Order 4 Length 6
7 \$ v* r# O0 }# U Rep (c, b, a, d)
8 U+ p5 ^ |: R
; l0 B( r4 `! r1 X" P
( a$ E: o* R/ Z& t6 B, O5 |
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发表于 2012-1-1 12:28
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