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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
9 U; ^: X7 \# B! D
# G! }1 D+ Z4 z' L/ D/ KS4 := Sym({ "a", "b", "c", "d" });* H! c9 O. m% N- X
> S4;
4 B) z- K# x1 {/ L) H8 RGenerators(S4);/ V# Z8 _9 t$ R% O. b7 u
IsAbelian(S4);不是交换群
' t5 t6 G3 D% V6 E# ?/ O. `- WSubgroups(S4: Al := "All") ;列出所有子群
1 f; i5 J6 i- m1 P Subgroups(S4: Al := "Maximal") ;列出所有极大子群& a, w w& u% u# Q. h1 E" W0 f) c
( ?& ~/ a# F5 g: x% j) qSubgroupClasses(S4);+ S6 ?. F. I7 N1 R% V
' Z) p8 k" h% |" F9 q- m# B
NormalSubgroups(S4);
9 u. z! i& S; ?$ q3 f" `AbelianSubgroups(S4) ;
' D" a9 Y h0 v) i/ n, D* J0 eMaximalSubgroups(S4) ;& r+ H2 d$ J! C# s" }- e
' r0 I }0 ?) d1 k: K0 c; k
SubgroupLattice(S4);成格，你可画下这群包扩子群的图) ^1 j: E E# [8 u p$ y
( c* l, ~% I2 z, q- `1 m
GSet(S4);
) O! S- W4 x n. c) uConjugacyClasses(S4);
3 @, Y& {1 S! HNumberOfClasses(S4) ; 5类" E/ n7 ?( A' D* U4 G; m, ~/ W4 R" c
+ U2 S/ O% R/ E1 h2 S5 L
Symmetric group S4 acting on a set of cardinality 4, L: k" R% W0 g, u, p+ T
Order = 24 = 2^3 * 35 B5 _% ]. y7 S1 Y
{8 r9 ^% Z0 D) O a7 n; d6 [
(c, b, a, d),- f! {" d3 E4 L6 z
(c, b)7 q8 O7 V# J) e' z
} 两生成元
7 |, @+ |3 n5 t/ Y5 k; v. efalse2 S Y2 D9 L$ N. H# M1 c
Conjugacy classes of subgroups 子群共扼类
0 T! U( R, e* W" Q7 G------------------------------: E1 G8 ]; h2 g* p9 k+ V! R8 W6 c
& t$ r( ]3 p* n) j% F( F$ w1 S
[ 1] Order 1 Length 18 `' Z; [5 F" m% M/ I. Q% z( a
Permutation group acting on a set of cardinality 4
0 h) d" H* D4 R) h0 J+ ?5 G Order = 1# n, m7 s- ?# |
[ 2] Order 2 Length 3- E" `) z) N; [" T2 Z5 _) ?
Permutation group acting on a set of cardinality 4
* b5 H; U% ~ s# t Order = 2+ [( G# \: o5 j0 k& u) Z& w
(c, d)(b, a)2 w! L+ }( h& k: O. S
[ 3] Order 2 Length 6
- @' \, l0 s* Y- ` Permutation group acting on a set of cardinality 4
! Z& ? U# U" }8 q K2 M5 \% d3 `7 L Order = 2
6 p/ {& g5 S u/ S; \ (a, d)- B2 u, W9 X: x
[ 4] Order 3 Length 47 T! j2 f! B3 X. O4 t/ b
Permutation group acting on a set of cardinality 43 J% r, a4 y8 {/ y8 q: h8 G7 N
Order = 33 ]" q! R- D/ F) O# m9 K
(b, a, d)
# ^8 r5 E: e5 `7 F[ 5] Order 4 Length 1
/ x7 ^: f. O c6 m. J5 A( M. l p0 g Permutation group acting on a set of cardinality 4; G/ R" L7 @( X% Q) b/ a
Order = 4 = 2^2
/ Y* P* G0 l7 v+ w (c, d)(b, a)' A x* k5 s2 D; k& V
(c, a)(b, d)9 n& i! ]7 o/ E$ q/ o
[ 6] Order 4 Length 3
1 T \0 j) v' Q7 y, a1 j8 ~7 d2 S Permutation group acting on a set of cardinality 4
2 f- W9 \% ~4 m. k- `& K Order = 4 = 2^2& g5 q5 O; ~! _ B
(c, d, b, a)
4 Z, D; Q5 a0 H6 G% t$ I( S$ |# h (c, b)(a, d)
$ ^4 }3 T9 r2 `) I5 b[ 7] Order 4 Length 3. T. o: V* U X' S# N( Y/ E
Permutation group acting on a set of cardinality 4
3 D, ]6 d/ {7 Z Order = 4 = 2^21 x) Z M# ] l" `. a
(a, d); T- S% g0 P$ N6 j- h
(c, b)(a, d)
9 y) `% q- k$ ?8 G4 c& P[ 8] Order 6 Length 4/ C3 i, J [/ w* m( w$ G1 v
Permutation group acting on a set of cardinality 4/ z) F; G. K& t: f
Order = 6 = 2 * 3' e1 z$ Y+ K/ \) ]
(a, d)' ^( K$ {/ \% `- Y$ J2 C+ @) J
(b, a, d). ], E! {7 Z3 H3 i; G
[ 9] Order 8 Length 3
2 X; p# L/ c5 O5 O1 L3 U3 r& K Permutation group acting on a set of cardinality 4
# _$ S+ E5 o) }4 c2 H. n8 l1 \ Order = 8 = 2^3
2 U* ?# j1 R! r: r (a, d): ]! P+ R5 k5 h# p2 V* T
(c, d)(b, a)
2 l" H- n, R! j: |+ C5 F& Y (c, a)(b, d)
7 ?8 t& e* S2 L$ R% X* c: q1 \, z; y[10] Order 12 Length 15 r; G2 M9 H, A: _, ?) T
Permutation group acting on a set of cardinality 4! P$ y$ O9 X/ t' I, A
Order = 12 = 2^2 * 3
- d" G; w- ^% n( E6 q (b, a, d)
4 M$ L- p6 l. _/ @- q3 \& [ (c, d)(b, a)
$ z, ~1 n- Z7 K, c4 k (c, a)(b, d)
2 X2 }: P! q2 t+ [8 q, K[11] Order 24 Length 12 [# D5 W2 L; ~! s; f9 O' }
Permutation group acting on a set of cardinality 4
& O ?% i6 Q ~3 S* M5 W Order = 24 = 2^3 * 3! G8 J- ^! x! B" @$ w1 }8 Y
(a, d)
1 G& f$ ~$ R. g( G (b, a, d)
* E; t. I$ c" Z( K, V5 g (c, d)(b, a)/ g$ `, H, P+ y/ z% q
(c, a)(b, d)
4 \+ o# n& |. z% E! l( CConjugacy classes of subgroups
( u8 O6 [/ P( Y1 u------------------------------& I1 G/ `1 g7 ~7 {- \ t5 x& ?
7 Q- e6 c: `+ Y3 z& h6 z: S[1] Order 6 Length 4
- |9 L2 @+ J( h; X+ A% {1 B6 W1 D Permutation group acting on a set of cardinality 4
. L3 G! A8 e$ w, s N+ d$ M9 W7 ` Order = 6 = 2 * 3
8 b. u, p. {5 p6 _& d (a, d)& h9 L4 t K" H
(b, a, d)
A5 X( C# G2 ]' C! x) A7 }[2] Order 8 Length 3/ o5 \) ~) G1 w2 Z" |$ H
Permutation group acting on a set of cardinality 41 s1 N# U1 ]% Y. K- z
Order = 8 = 2^3- L2 M8 X, |0 Y2 H
(a, d)2 }2 W1 h) Y* b0 }6 B( U" N
(c, d)(b, a), c0 V7 T) ?$ r
(c, a)(b, d)
9 j( p% Q' M. c0 j3 R[3] Order 12 Length 1* n a* M8 R% K* ^ b
Permutation group acting on a set of cardinality 4& \1 q8 ~8 A/ l- |! D: C3 N M1 S
Order = 12 = 2^2 * 3
. m1 @+ l& _, y: ]8 H (b, a, d)) y5 s. a* h. T& ]5 u
(c, d)(b, a)
6 ^+ O9 w' C1 d (c, a)(b, d)) @$ Q' u* u/ Q# N
Conjugacy classes of subgroups
, u- X' W4 H1 E) _) ^) d, O) V------------------------------* j1 W, W% ~0 x8 I
( Q- W, F; J* X O( {[ 1] Order 1 Length 12 ^( {/ G' x- \" U3 f; i' K6 o
Permutation group acting on a set of cardinality 49 F1 E4 W' F) [: n" F/ Y. W
Order = 1* |3 s( _) M. ]
[ 2] Order 2 Length 3
- o9 g! s6 J$ b4 @6 \6 g Permutation group acting on a set of cardinality 4
$ Y* C# ?7 q. @ Order = 2& h- ~4 a6 Y {) m8 c- v7 p0 S
(c, d)(b, a)
7 H7 c# s/ C/ i[ 3] Order 2 Length 6( L. W% z- @- G$ s3 O; w* X
Permutation group acting on a set of cardinality 4( q* S: y8 j. H% E0 x" o5 Q9 a
Order = 2
! T: ], B4 Y% ]2 G0 `0 R (a, d)5 K3 t) {& A: x% b% z7 K5 p& O
[ 4] Order 3 Length 4
5 g& |4 [* U* H% b Permutation group acting on a set of cardinality 4( z* O. j+ d+ k9 v- N" M1 T
Order = 3% S: O6 n5 a7 Q! T$ x
(b, a, d)
. A& V" k8 \" {5 p/ D[ 5] Order 4 Length 10 t7 q0 e. h( s) T
Permutation group acting on a set of cardinality 4
# @- s+ ~9 F0 b- ?- A0 J7 s8 A Order = 4 = 2^2
5 {( T& _: h: N& S) a (c, d)(b, a)
! Y+ u7 J, c2 e* r (c, a)(b, d)
$ G: k6 t @) D6 v2 h. V[ 6] Order 4 Length 3
* `3 n! h! h; k Permutation group acting on a set of cardinality 4
# g. {$ z, S! R/ h# }. u Order = 4 = 2^2
6 r5 [ k& s: ^ i (c, d, b, a)) A) _7 m. t5 C
(c, b)(a, d)
, G- |/ F8 I% P: v9 B6 e[ 7] Order 4 Length 3
7 U1 F" ~* D9 a7 j* _ Permutation group acting on a set of cardinality 4
7 x5 Y+ w! h/ |, A, @3 U+ |8 W Order = 4 = 2^2
3 k, n, P: `: a2 u! J (a, d)
5 o: c G3 ?' ?4 H (c, b)(a, d)& O! E; G+ i& q1 O0 T- o& d
[ 8] Order 6 Length 4, d0 k* }7 C1 W ^) Z6 H. P# O
Permutation group acting on a set of cardinality 45 Z$ C" ~/ u9 ^5 A, _ |
Order = 6 = 2 * 3$ M/ ]8 A6 N% Z1 d
(a, d)+ x5 W3 n% V* W; r' f
(b, a, d)
* m5 G8 G( }, ^4 Q[ 9] Order 8 Length 3
( D2 v. U: _+ A) [/ Z; Y Permutation group acting on a set of cardinality 4/ S4 v# ~5 X# v" Q$ q- D
Order = 8 = 2^32 Q' e) v6 F9 l7 N0 B5 {
(a, d)0 ^% k/ |! Y! ^. F( {
(c, d)(b, a)" y O: {' } V
(c, a)(b, d)
6 v4 _0 v# z- d0 d- R[10] Order 12 Length 1 z1 C. g u5 B. X
Permutation group acting on a set of cardinality 42 Q. ~% G' i, `; l+ P4 b
Order = 12 = 2^2 * 3
" Q( L- N2 w& |" _0 b( C (b, a, d)
8 P" A" ], h r1 g (c, d)(b, a)
) t& H( E! a8 D w- z p" z (c, a)(b, d)
/ U- d! A8 B7 o+ l, u* S3 T1 S% ?[11] Order 24 Length 1
4 |. p3 L/ K/ K Permutation group acting on a set of cardinality 4" d$ l% p7 K$ H+ d6 V0 E* H
Order = 24 = 2^3 * 3
& P" R0 W7 |- [" z1 U5 R* \ (a, d)+ n+ k5 l0 N2 c; `# }
(b, a, d)
8 _ `8 {8 `$ g2 p9 d& L2 r% ^ (c, d)(b, a)
, k$ Q: F8 @ Q& y; n( H, B (c, a)(b, d)
( s+ h* L/ }; G. g UConjugacy classes of subgroups `8 l7 e+ T/ H6 _) f# s* d
------------------------------3 b# H! ?) r# m+ C; ^$ j# @
. f1 z' {4 V# J& U, K# I0 |7 D* r[1] Order 1 Length 1
% C% R; k; |5 l Permutation group acting on a set of cardinality 4
5 @: R: T2 B7 F( K+ k( D0 a Order = 1+ v: F b: \/ G
[2] Order 4 Length 17 d r3 P7 E( G( F0 P* Z, A; ^: [& }
Permutation group acting on a set of cardinality 4
: F% P- N. [8 \ Order = 4 = 2^27 G7 x1 T5 M* Z% V$ F% C) j; P
(c, d)(b, a)) |8 U- v; \3 L" q5 O6 x
(c, a)(b, d)4 m4 m+ h! J* o9 I
[3] Order 12 Length 1
9 M3 q7 v$ Y) m Permutation group acting on a set of cardinality 4 }9 T8 M1 R. {6 l
Order = 12 = 2^2 * 3- w1 b7 M/ q4 _ @* D
(b, a, d)
9 l0 j& \4 ^& u (c, d)(b, a)
3 _+ w& |6 N$ h. ]3 v( w (c, a)(b, d)& n+ ]- t: r. p8 \* t* s' N
[4] Order 24 Length 1" _& d* ~+ N3 l/ N; V# h
Permutation group acting on a set of cardinality 4
9 n# N3 j9 H( @" G5 @6 |/ V! R/ q Order = 24 = 2^3 * 3
" W7 E$ B; r8 Y6 ?' R2 K (a, d)+ I. {. g0 y# w2 f
(b, a, d)! i* o5 P- e) F# T" d' l8 P
(c, d)(b, a); c: F6 a# X4 v& V5 A; y
(c, a)(b, d)
* g: k" D# W* ]$ Z; ?4 fConjugacy classes of subgroups6 m+ p e) }; [; P# G. A
------------------------------
! v# S0 }( c/ ~/ V: c( M* Z8 }& C
# f" w0 L1 p$ b$ Z[1] Order 1 Length 1
) _( n# @+ d! f; I5 b( L3 X Permutation group acting on a set of cardinality 4% Q9 G! _" [) y
Order = 1
9 T- G) [( D5 Q& w: u[2] Order 2 Length 3
$ o2 s5 |+ C) a& v( W7 v5 k Permutation group acting on a set of cardinality 4+ `6 S# H1 A8 r* N' [* p% o
Order = 25 t" i% o/ F* o) q
(c, d)(b, a)) j- l. [, o7 ^( e/ D( p
[3] Order 2 Length 6
{* u7 n' ?5 F* v5 g0 k( s Permutation group acting on a set of cardinality 4" V% F; _" C0 _5 t* q* B! l
Order = 2
* A; v, d8 H5 z6 t1 P (a, d)
! }/ {% ~3 P2 x[4] Order 3 Length 4& Y$ x( p, n7 Q/ Q8 g
Permutation group acting on a set of cardinality 4
% w( F6 |5 L7 h6 J: W+ O Order = 3
' n/ ?9 _% Y8 [5 X (b, a, d)
$ P8 i6 ~5 U! s1 ~' u[5] Order 4 Length 1" {3 M; Y F b
Permutation group acting on a set of cardinality 4
6 W5 J5 Q# }2 P7 R# Q1 _ Order = 4 = 2^2 t. q/ r' w# f5 i% K3 t. `+ S! e
(c, d)(b, a)
2 p6 Q: M, s1 ]1 z j$ Q+ ]0 V (c, a)(b, d)4 [5 X1 v4 W7 [' m% m$ }* H
[6] Order 4 Length 3
- ?7 ~1 {+ a) j, b Permutation group acting on a set of cardinality 4) r0 @+ P: G+ ~
Order = 4 = 2^2" j9 b' r% `6 ]3 o" O
(c, d, b, a)6 ?) |4 ~$ S) ^* k' k3 @- [2 g
(c, b)(a, d)
; B' E5 z& Z3 K[7] Order 4 Length 3/ L/ |* ?2 [# _ c# U( t/ m
Permutation group acting on a set of cardinality 4
& c* b0 c3 w* i, r6 \* a4 d1 I5 f$ ] Order = 4 = 2^2
, c# d8 D& X" D# ]) D (a, d)
) t9 ]8 b7 `9 a& E (c, b)(a, d)5 K/ G" O) R5 ~
Conjugacy classes of subgroups( x5 C5 j3 Y: d* c
------------------------------
7 | O' V/ U i! Q' Y
6 R; N/ V0 Z4 R# l[1] Order 6 Length 4
" i! f; S9 }) e5 G. {9 k Permutation group acting on a set of cardinality 4- u( }$ A5 c! i* x5 u
Order = 6 = 2 * 3
% H+ |* j4 c% i$ l" v5 o (a, d)- i# Y) N d `
(b, a, d): l6 e( m0 r" f, Y( J" f D5 g
[2] Order 8 Length 3
/ O( A4 o3 Q h# \ Permutation group acting on a set of cardinality 4
6 f- c. f' M4 t' C' J Order = 8 = 2^3
' B- u2 P T& K) \6 ~6 e7 @8 X (a, d)
+ I* o0 M$ e: a3 j" u/ q (c, d)(b, a)5 `% j9 q) V6 \& e0 Y$ D
(c, a)(b, d)
1 h: b z, Q- S' `/ v[3] Order 12 Length 1
8 t( Z" o3 A1 g. I# e8 I8 M6 U$ Q Permutation group acting on a set of cardinality 4/ Q; S+ j: b& b7 \
Order = 12 = 2^2 * 3$ n# I2 K; l: N: ]0 V# J& F2 Q
(b, a, d)) k) v& P) t( O
(c, d)(b, a)
" N5 B7 E& Z2 L' q (c, a)(b, d)
, [0 Z& x$ U4 g% n% F/ Z J# y K, S3 r7 f, e4 ^; j& B( V
Partially ordered set of subgroup classes
4 r. n: L1 t. }, R" ]$ a+ B-----------------------------------------0 W( o* g5 h N3 E: E
) x3 A: g3 [7 X2 R[11] Order 24 Length 1 Maximal Subgroups: 8 9 103 y% g$ J- K, E) b$ P9 [
---
$ o0 r0 r! r- ^, u& U[10] Order 12 Length 1 Maximal Subgroups: 4 5! c2 |4 B7 B J
[ 9] Order 8 Length 3 Maximal Subgroups: 5 6 7
) w$ K8 M+ M8 e, e$ N4 H; ]9 m7 {7 |---; H$ A* |2 [: c; g8 r L4 ^, J
[ 8] Order 6 Length 4 Maximal Subgroups: 3 42 p: m2 K7 u: X) m% k# q
[ 7] Order 4 Length 3 Maximal Subgroups: 2/ I4 e# l+ y- Z& Y6 H' t! l
[ 6] Order 4 Length 3 Maximal Subgroups: 2 37 ]' n' j/ ?! w0 L
[ 5] Order 4 Length 1 Maximal Subgroups: 26 j' m0 T6 ~5 p. s* l4 H1 {! v
---5 ?8 n" E7 f# o) i0 |2 T; D
[ 4] Order 3 Length 4 Maximal Subgroups: 1
+ d! A% R0 G* n8 d' h3 Q[ 3] Order 2 Length 6 Maximal Subgroups: 1
2 A/ Z: g: _6 J2 A, G& W. J[ 2] Order 2 Length 3 Maximal Subgroups: 1
- }% [2 w+ w3 }. B% I---
6 ]+ n7 J& V2 V+ y[ 1] Order 1 Length 1 Maximal Subgroups:& j; l7 q: A9 T3 M a; W( c, t9 S( R
( {) L! C. b( Z( N& j1 u
GSet{@ c, b, a, d @}
9 C! C& c+ c% t1 m" a9 P7 a1 nConjugacy Classes of group S4
; A6 x n4 ^! {-----------------------------
! w8 v0 ]0 Q2 f ?# U% ]4 g[1] Order 1 Length 1 3 R) W3 p, z* K
Rep Id(S4): h& r( L3 |4 ?/ G; O9 i
+ q9 V" y: Y8 W/ c: ?( x5 k[2] Order 2 Length 3
; x! @: J+ p4 n8 Z* k+ h Rep (c, b)(a, d)
& E, A9 U' l6 n) W4 C: {" a7 }% t3 d6 ]3 `8 W) |+ J# P' V
[3] Order 2 Length 6
; ?+ `1 k6 |. X O4 e8 f& M* R Rep (c, b)) Y( g9 Q4 c& s& F
0 O+ _% ~ `6 K0 a0 h, i7 C* Q
[4] Order 3 Length 8 & U$ O9 q4 m8 `5 U6 }* x
Rep (c, b, a)
- M1 ~2 J9 |; A9 r7 G1 ^( Z* t" R3 ~- S E5 P" z
[5] Order 4 Length 6
# Q2 ?1 |3 L( f0 B3 f Rep (c, b, a, d)
$ i! W* u* @4 S1 t& b4 P
- c- p( C! K1 r* o: y& D; C8 p3 w4 e1 a) Y9 r7 Z% O
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IP卡
狗仔卡
发表于 2012-1-1 12:28
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