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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! c" i# M% G( [1 E) _( a
1 l/ F5 _+ u% j: v2 P1 g RS4 := Sym({ "a", "b", "c", "d" });
# s) l7 J+ C4 e$ x8 v! Y> S4;
2 m" ]$ u4 u4 u) kGenerators(S4);
. W$ ^) b3 @; V/ Q0 y: fIsAbelian(S4);不是交换群
1 q% o/ N; w0 _+ T" _% w7 zSubgroups(S4: Al := "All") ;列出所有子群
# x) n8 N z! r8 F! _2 z Subgroups(S4: Al := "Maximal") ;列出所有极大子群
n( o& f: d9 v) M5 G; }
5 ]: _, y8 ^( ]/ ?SubgroupClasses(S4);- C+ }3 X; s2 {, S% p- z
3 n+ _$ k' v$ n# M M, ~
NormalSubgroups(S4);
) J* y2 j4 O* P1 K% j7 ?AbelianSubgroups(S4) ;
/ Z7 K9 ]5 I7 n, ]5 B7 a0 ^MaximalSubgroups(S4) ;
5 D/ T5 R& x/ s( ^6 G2 l
: e! _1 k# o! x" ]! {0 z' JSubgroupLattice(S4);成格，你可画下这群包扩子群的图8 |3 O. ^; W* \) F7 C
, t5 r; E: h' d0 z/ ?" l% ?
GSet(S4);
" l# t7 w" i$ g6 cConjugacyClasses(S4);
, t6 k8 G+ C( ^# ONumberOfClasses(S4) ; 5类+ d" T0 V7 \. d8 d
7 L; s7 u5 D- H% q- ~
Symmetric group S4 acting on a set of cardinality 4
3 f. o1 S0 k. n5 rOrder = 24 = 2^3 * 3
: h; A8 d/ g5 x2 Q4 @5 V{
5 z) j9 ]7 S% w+ ~9 v (c, b, a, d)," y" g$ k# k$ s* j4 E
(c, b)% o9 K$ |/ x% j7 T6 T$ \
} 两生成元1 `# ?) F9 g2 w* }
false
5 v; Z1 B7 N N/ x# e7 ~2 DConjugacy classes of subgroups 子群共扼类
1 N: e& d- o$ ~- S4 U6 H6 w------------------------------
' w# x( v1 [4 }. m' m4 u: q2 o
/ O+ v7 Z4 p% q8 s! g" ?% W[ 1] Order 1 Length 1
' q \" m' M+ W- f Permutation group acting on a set of cardinality 4! \3 u! X1 E- S/ l6 K6 D' Z
Order = 1
A5 P" s# U s, i9 G' h[ 2] Order 2 Length 34 j+ P3 w- ^$ W& H p' A& \
Permutation group acting on a set of cardinality 4/ g$ R: F- Y( v0 K4 r
Order = 2
% ?; n7 K7 S. B3 L a O$ t (c, d)(b, a)( @) f/ t3 I8 U+ k
[ 3] Order 2 Length 69 ` }3 U* K1 B6 \& ?. W' A
Permutation group acting on a set of cardinality 4
% X) u$ e$ L/ U/ K Order = 2
J2 R$ p! Z9 N; c- X" G, {! K2 q3 L* q (a, d)
D& W8 {4 A) Y5 t. w. H- _" z[ 4] Order 3 Length 4
4 x0 ?( p( s! D: }4 h+ D Permutation group acting on a set of cardinality 4 s& l w5 _* m$ A; V
Order = 3$ M6 X9 V, }; j
(b, a, d)
: w8 V; q) d, {$ m. [! i- J& L[ 5] Order 4 Length 1
o! j5 B8 i% K/ q Permutation group acting on a set of cardinality 4. ^- L# I1 m* j- n
Order = 4 = 2^2
$ u, T; b, A1 I (c, d)(b, a)( o3 [1 O/ }! I% s( ^" r
(c, a)(b, d)6 I2 {3 o" `! C6 Y/ F) B& p
[ 6] Order 4 Length 3
, S; O7 P4 e5 K0 E0 k% Q N) J- i F8 ? Permutation group acting on a set of cardinality 4
, G2 l* Z7 d7 D; `: M Order = 4 = 2^2
, B& w7 D0 e- Q5 }! h, d( Z (c, d, b, a), y; n- ?/ ]9 }: w
(c, b)(a, d)
. L- _2 [4 E" w[ 7] Order 4 Length 3# @7 {/ ~1 ~, ]. b
Permutation group acting on a set of cardinality 4
5 a5 V5 \* i% P4 v, R1 u9 u Order = 4 = 2^2
( l1 e L" B8 p6 f6 P' F- r- | r (a, d)3 ?8 U6 k4 K. `; p
(c, b)(a, d)* {0 z- O9 A# t* W: V+ K
[ 8] Order 6 Length 4
4 `) \) W- w5 k* ]0 ~6 c4 H! Q Permutation group acting on a set of cardinality 4
+ Z* c8 e& w# A+ |" y' ^* v, q; k Order = 6 = 2 * 36 A" L' G! A! \( T
(a, d)
, ?; s+ R' ^$ |+ ^0 F( o2 A (b, a, d)
' h: w" Y! [8 G% r7 ?0 v# J4 ?[ 9] Order 8 Length 3
! Y2 |7 n- I, v Permutation group acting on a set of cardinality 46 y3 C" ?8 _: l& R" J8 Q9 [8 `( A
Order = 8 = 2^3
/ {3 j- n. k6 R5 B: { Z# y. c (a, d)& ?% z2 C2 Y) @; U% ]
(c, d)(b, a)' K# ]. P8 q$ k8 \9 i3 p' \% Z
(c, a)(b, d)
- t. g1 T6 n" N$ E7 q* u[10] Order 12 Length 1
' B: |: ^8 r, o: H Permutation group acting on a set of cardinality 45 ~5 ]8 _+ u7 b. u4 R# ^, ]
Order = 12 = 2^2 * 3
" ?- K" v( n& T+ X8 j; @3 ~' f (b, a, d)0 [1 S6 ^ v" F+ S- {, Q) Z! n
(c, d)(b, a)
; ?: {3 h: L4 { E" c: r (c, a)(b, d)* d5 K* ? b2 h1 e
[11] Order 24 Length 18 }5 ^. L" R5 E' c0 Q! n
Permutation group acting on a set of cardinality 41 @5 U% |+ i* i% D) u# a. N
Order = 24 = 2^3 * 33 d# s* C7 J' s
(a, d)# O5 p: b* {8 x. E7 N
(b, a, d)
$ y) r$ Q0 C; @ (c, d)(b, a)/ C: u& S- D1 X: B* _5 U
(c, a)(b, d)
+ `# K6 D* }2 n" f$ h) D- SConjugacy classes of subgroups
: N, t7 H; [1 Q" O% Q4 \. }' K------------------------------
8 \& f2 i+ ^ t6 u# d0 E
9 R, L5 X& M0 g7 G1 F, L. s[1] Order 6 Length 4
. [& m1 l8 N# A& Y Permutation group acting on a set of cardinality 4: B6 L: ^. \6 f+ U ~( h* y
Order = 6 = 2 * 3
# P0 S; F; b, s! J+ f" r4 n (a, d)* \, i8 Q2 U$ J! h! j1 D
(b, a, d)# S4 _7 v) _' L& r! Q- ]
[2] Order 8 Length 3" F1 S7 a# j3 U1 W
Permutation group acting on a set of cardinality 4
2 k/ a" J. `3 H/ |& F5 ^- i2 q Order = 8 = 2^3
% \6 l T/ L( V% K- S (a, d)
( {. r3 |+ l2 N Q% F& ` (c, d)(b, a)! j* x# c; `. J
(c, a)(b, d)0 P, h( u5 m- X: y( d
[3] Order 12 Length 1
[6 k" Y2 c* k! v# i Permutation group acting on a set of cardinality 4! `4 y2 A+ _' T
Order = 12 = 2^2 * 3
0 b+ B; D) q" c5 I (b, a, d)
, v. k9 ^+ d) U6 ]7 _ (c, d)(b, a)
5 @8 _$ z( S% w$ G5 b (c, a)(b, d)( [9 J, R& V! I/ F
Conjugacy classes of subgroups
' ~" s3 I0 p8 G9 N2 D4 y------------------------------
! x& p2 k+ R% L2 |7 E) F0 p/ L1 D. u% Y/ Z
[ 1] Order 1 Length 1
6 Q' }* Z) F6 Z Permutation group acting on a set of cardinality 4* J' k* s+ w" d& h
Order = 11 f+ N" Y5 q! G6 @# P. N; {
[ 2] Order 2 Length 3
; u) y+ L8 v, K8 I/ v Permutation group acting on a set of cardinality 4! i* J. H; s! m8 U7 {5 O6 v5 q- G$ y
Order = 2
s' E8 q* R) U) n' b (c, d)(b, a)+ t6 K* s3 I' N4 l- k+ f9 e# R
[ 3] Order 2 Length 6
$ g+ u- i0 M& K# x Permutation group acting on a set of cardinality 4
+ G# O1 e6 m0 }7 `/ |( O! ] Order = 2 u7 B4 ^- q" b. P6 s
(a, d)/ f! y( {% R, P( H8 Q
[ 4] Order 3 Length 4
3 H+ m+ m! O9 V5 H d Permutation group acting on a set of cardinality 4
" p1 [/ K' H+ }: D" R, Y Order = 3
: L+ A# `# E$ m( N (b, a, d)
) Q; O( l6 h4 w[ 5] Order 4 Length 1 v2 S3 g! Y: S% j. p
Permutation group acting on a set of cardinality 4
, L5 O1 d% d% V& e! [7 a# y+ K- ] Order = 4 = 2^2
5 [. x! {: c0 ]" Q: Z (c, d)(b, a)
1 I' \. I4 u& Q" y! E7 B' V$ O- k) ] (c, a)(b, d)
6 p/ B% @1 R \: o4 u4 q# l# w[ 6] Order 4 Length 3
3 y& n& l7 ]& x# Y2 {0 p Permutation group acting on a set of cardinality 4
" ?8 F4 X% ]1 l( S( A Order = 4 = 2^2
' q4 {8 k5 p+ u+ T2 w8 y (c, d, b, a)
! H8 d0 h% q i8 E% {! t (c, b)(a, d)
2 k4 P' d7 b J4 b% Q- f9 T[ 7] Order 4 Length 3
f$ s, {5 }# ^ Permutation group acting on a set of cardinality 43 [# \2 w$ {: c8 o
Order = 4 = 2^20 I' }% x( z6 h' w, r$ j! Q4 a
(a, d); [* Y' G% p6 |$ i( Z/ |
(c, b)(a, d)1 y5 s' u$ K7 _
[ 8] Order 6 Length 47 J. W) t2 }$ f
Permutation group acting on a set of cardinality 4( T0 N! a0 u8 v2 Q$ r. e
Order = 6 = 2 * 35 M4 J1 _7 A" T
(a, d)
2 q! o! f( C5 y8 V (b, a, d)
& x) L$ ]0 g- l) V, {! H+ u7 ~[ 9] Order 8 Length 3
/ Z5 B0 j) m$ p: }* z Permutation group acting on a set of cardinality 41 |6 S/ f3 r' i* Z$ @
Order = 8 = 2^3( L* k/ b8 V$ h' K5 u" s
(a, d)& W; m$ P ~ w8 s& d$ C
(c, d)(b, a)! f- H! V7 ~& u" W
(c, a)(b, d)2 j. n0 Z) Y( }! E
[10] Order 12 Length 1
1 O& L1 V* l6 [1 L# S, _ Permutation group acting on a set of cardinality 4
+ H, [$ }0 x! L. N: ~4 U0 f Order = 12 = 2^2 * 36 T/ X, f, L" L
(b, a, d)
( e6 o; g k' n, T) P (c, d)(b, a)
) a D+ ?: i. ^6 C4 ~$ E4 d (c, a)(b, d)
4 F! u8 j0 E7 C1 ][11] Order 24 Length 1
+ S6 ]8 p8 ]5 k- `) j6 R. v Permutation group acting on a set of cardinality 44 m* ~1 F3 T' ?! k' Z+ u
Order = 24 = 2^3 * 3
7 ]/ F P: Z8 ]7 s# L- o; Q (a, d)
& z! O9 i% c/ E }0 C (b, a, d)6 ?7 a( @2 n- C O- O$ O0 v: Y. e
(c, d)(b, a)! g T" t# j* J9 C
(c, a)(b, d)
f t# Q" [* B0 n4 { ?0 ^ RConjugacy classes of subgroups4 x3 m0 I5 U/ {* ^/ w. |
------------------------------$ E0 u4 K" @9 u& v
- b6 Q' k4 r5 U; N7 ^* d
[1] Order 1 Length 1
3 f2 o# A2 j' }6 s) p Permutation group acting on a set of cardinality 4. q( H/ U1 ^1 B9 l8 m, m
Order = 1* H* U! q! `" U% i6 u' h. I
[2] Order 4 Length 1
, l; s' ~ _' ?+ F3 O' | E* a Permutation group acting on a set of cardinality 4
6 U; f7 p; p; C- U+ d; ?6 f% v Order = 4 = 2^2
: M. H& z8 q; q7 q6 w1 p (c, d)(b, a)
9 @) g6 D0 `0 _. a# Z (c, a)(b, d) ?( \. o V( G# y
[3] Order 12 Length 1# e' H' i8 E! Y7 C8 F8 Z
Permutation group acting on a set of cardinality 4
' p' R; y# U+ _0 ~ Order = 12 = 2^2 * 3; B) D- B: V/ M6 \
(b, a, d) N: J( }7 @2 I y1 A7 O4 j
(c, d)(b, a)+ X) t8 Z; K; W9 T
(c, a)(b, d); q4 @2 ~# F7 q( q R
[4] Order 24 Length 1
# a0 m' N. ^6 ]5 S4 I+ E3 y) T' B Permutation group acting on a set of cardinality 40 T1 g0 Z* t' L5 K" F" l/ m
Order = 24 = 2^3 * 3
- H s3 D6 |+ L (a, d): Y4 ^5 y; E; z" b' r' u/ \
(b, a, d)
3 S3 L9 [8 m/ Z# O7 }" | (c, d)(b, a)
/ \; k5 q/ ]5 |8 d; B, V' _ (c, a)(b, d)- \9 M9 f- Y. ^( u/ v# N
Conjugacy classes of subgroups- w' I9 _4 a1 t
------------------------------3 W4 l3 M0 V# V4 @. {
& z" P0 C( c- O# w, `& d& y6 Q
[1] Order 1 Length 10 [* w& v- y2 f* f8 y, w" b3 Z
Permutation group acting on a set of cardinality 4
+ U6 i! P9 J2 X3 P* N& g F; ] Order = 1
3 K( Y6 Y8 q s[2] Order 2 Length 3" J! z7 S! d# I4 [- ~, n
Permutation group acting on a set of cardinality 4
9 p6 g# P. B1 q Order = 2! Q: V/ w2 d( {6 x& v/ _2 V- ^
(c, d)(b, a)
+ L4 ~$ N# |4 q- w; J x, A[3] Order 2 Length 64 w5 F9 [" R% e" ~
Permutation group acting on a set of cardinality 4& W9 W [) d) y' t. T) h
Order = 2+ b0 c: }2 S6 b. O
(a, d)
& o3 |8 N6 i# m {, z: b[4] Order 3 Length 4
1 U3 {% e5 j* x C+ }0 ` Permutation group acting on a set of cardinality 4
+ r' n6 @& C2 a( R5 q3 n- L" W Order = 3
( `) }: Y- r# R4 ?6 S4 I. i (b, a, d)2 z8 q( [! F9 u* g6 Z0 Y. d0 ~
[5] Order 4 Length 1
/ G# K/ i; `$ b Permutation group acting on a set of cardinality 4* @( @4 @* g1 t O; {" y8 e! u
Order = 4 = 2^2
# E8 Q9 }: q4 \ (c, d)(b, a)
' L5 Q& E1 y" G: L" d" d (c, a)(b, d)) |% f$ u2 L8 Y+ z
[6] Order 4 Length 3& S6 U# k$ C/ a% }8 c# O4 M% C
Permutation group acting on a set of cardinality 4
3 S+ @& w4 Q* O Order = 4 = 2^2
/ p! F7 o- b' V$ m1 n- z8 @ (c, d, b, a)
$ ^! B- i3 J; q* f. T& z (c, b)(a, d)
4 |* o- ~, w8 I8 B: g0 G7 I[7] Order 4 Length 3
7 \. L8 ? g+ P$ O x Permutation group acting on a set of cardinality 4
. B" G5 o5 C# u/ B Order = 4 = 2^20 p* Z4 n+ s: s$ Y1 T! d* w4 {
(a, d)
! [# j) Y: u! k" z1 J (c, b)(a, d)
# D+ M8 Z. s* U( lConjugacy classes of subgroups
( y& g S9 R) R* J------------------------------, o6 n. {2 ?& C$ R% K8 Z5 l
, s# r' A+ B" Z[1] Order 6 Length 47 s$ ?6 Y" f9 q7 K5 ]' U
Permutation group acting on a set of cardinality 4
8 o5 F8 n4 A& R6 L& K! W" A2 F6 ]6 D Order = 6 = 2 * 3 d, r8 ^: | W) f7 e# n/ @
(a, d)
/ c4 R5 n# U" t (b, a, d) n8 q* ?! v: }8 z# v9 n9 P2 A+ A- T' d
[2] Order 8 Length 3. x- p/ k5 y: F I R8 V% A0 G) Q4 o
Permutation group acting on a set of cardinality 4) \: [: F1 w) Z1 B; e
Order = 8 = 2^3* v- z7 n* n5 |
(a, d)2 ?( v% v8 s/ R4 J+ o
(c, d)(b, a)6 P9 |% t: U* N* {
(c, a)(b, d)* B0 \) N5 S8 l
[3] Order 12 Length 1
, H3 p$ z. Z+ w3 h1 B& o Permutation group acting on a set of cardinality 4# M- v* i0 J1 u3 l
Order = 12 = 2^2 * 3" k {; V: i, t8 I( X
(b, a, d)3 u3 m$ ]5 n8 h4 W& R _/ h
(c, d)(b, a)
+ m' }- b G0 z }+ |. V8 \ (c, a)(b, d)* U# V' l a! w; V! H. E5 }; e
7 h1 p# K2 G( ^8 J+ j3 ?) }Partially ordered set of subgroup classes* I% }# e9 m6 @2 V( ~$ C) k
-----------------------------------------
/ A2 y) ~0 g9 W' U& P! d' D7 G
1 i4 X/ C' m! T$ N[11] Order 24 Length 1 Maximal Subgroups: 8 9 10
5 {7 V8 c: {3 ?9 w- B) K( U; C2 S: P---
3 h* \8 r7 V$ F[10] Order 12 Length 1 Maximal Subgroups: 4 5
# N) V. E7 i- z& z8 ^. F4 k- K" r4 k[ 9] Order 8 Length 3 Maximal Subgroups: 5 6 7
+ R/ i; `9 N M$ {; B. g---
! C. S& B$ x2 b' V3 d[ 8] Order 6 Length 4 Maximal Subgroups: 3 41 r+ x) ?1 E, u9 h7 l. ~
[ 7] Order 4 Length 3 Maximal Subgroups: 2
8 K+ L2 R& h* ^" n: [) S0 E5 W. N" V[ 6] Order 4 Length 3 Maximal Subgroups: 2 3
0 L# c+ C! q% q/ C* \& s4 J. H[ 5] Order 4 Length 1 Maximal Subgroups: 2% X8 K, j+ H" y* h* R; H
---; q. W! ^1 _' w! z+ [
[ 4] Order 3 Length 4 Maximal Subgroups: 1& N( x6 s; y# u
[ 3] Order 2 Length 6 Maximal Subgroups: 1$ l' {9 }) K- ^& S9 D
[ 2] Order 2 Length 3 Maximal Subgroups: 1+ U2 X% X3 j1 A- H+ |2 _1 [
--- E; v+ t" L4 V+ C% i
[ 1] Order 1 Length 1 Maximal Subgroups:
& I1 A7 F) e) H
3 y+ t0 ]/ B4 y3 |/ H vGSet{@ c, b, a, d @}# S7 n: M) J4 f( Y& F( v7 @
Conjugacy Classes of group S4, ~; w. G" l; g# R9 g
-----------------------------
& ^8 Y4 Y9 r) t$ t/ I8 N0 w0 k& j[1] Order 1 Length 1
$ y/ f& U/ Y; r h# ]' y0 ?8 [: e0 F# a; a Rep Id(S4)
" K4 y- q9 |. K) c/ ~- N% A% D' H4 h+ K
[2] Order 2 Length 3
L! H, J1 K1 a6 ~' H Rep (c, b)(a, d)
4 B, l4 F- W) m# _* {
7 k5 [0 Y7 ?, @" X[3] Order 2 Length 6
7 y9 a" A+ l! k' E9 ~9 P Rep (c, b)
; [; g' R6 [! O# B5 c! Y+ {9 L
3 P% O9 B% c! L[4] Order 3 Length 8
6 w5 L1 V x r8 t3 O% a Rep (c, b, a): h6 f& P& p' U
+ |5 m# X7 \! z' F
[5] Order 4 Length 6 1 L* Z9 z# y% ]# b9 i
Rep (c, b, a, d)
: O3 c! d \3 T" Q, l' K b3 Q5 y1 @) w8 } y
7 r2 j( n/ U. V. ?7 D6 J
5 |
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发表于 2011-12-29 00:36
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