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[推荐]近世代数第1讲

发表于: 2011-4-12 19:23 4178

[推荐]近世代数第1讲

lilianjie

2011-4-12 19:23

4178

当然不是我来讲，可谁有漂亮点的女老师的也贴上来

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cjsh88：

徐振环编的《群论导引》在线阅读。 ... 群论导引. / 徐振环编 / 1985年02月第1版
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群类论
8a6K9s2c8@1M7q4)9K6b7g2)9J5c8W2)9J5c8Y4y4Q4x3X3g2U0K9r3q4G2P5r3W2F1k6#2)9J5k6h3y4G2L8g2)9J5c8X3g2T1L8$3!0C8i4K6u0r3L8r3W2K6N6q4)9#2k6U0t1H3x3e0l9#2x3o6t1H3i4K6u0W2K9s2c8E0L8l9`.`.

西师版-张广祥-数学与应用数学-近世代数第2讲
西师版-张广祥-数学与应用数学-近世代数第3讲
西师版-张广祥-数学与应用数学-近世代数第4讲
西师版-张广祥-数学与应用数学-近世代数第5讲
西师版-张广祥-数学与应用数学-近世代数第6讲
西师版-张广祥-数学与应用数学-近世代数第7讲
西师版-张广祥-数学与应用数学-近世代数第8讲

求S4/S3凯莱表：
z1:=IntegerRing(6) ;
z6:=AdditiveGroup(z1);
z6;
AutomorphismGroup(z6);
Subgroups(z6);
IsNilpotent(z6);
G := DihedralGroup(GrpPerm, 3);
G;
f := NumberingMap(G);
f;
[ [ f(x*y) : y in G ] : x in G ];

Abelian Group isomorphic to Z/6
Defined on 1 generator
Relations:
6*z6.1 = 0
A group of automorphisms of Abelian Group isomorphic to Z/6
Defined on 1 generator
Relations:
6*z6.1 = 0
Generators:
Automorphism of Abelian Group isomorphic to Z/6
Defined on 1 generator
Relations:
      6*z6.1 = 0 which maps:
      3*z6.1 |--> 3*z6.1
      2*z6.1 |--> 4*z6.1
Conjugacy classes of subgroups
------------------------------

[1]    Order 6          Length 1
      Abelian Group isomorphic to Z/6
      Defined on 1 generator
      Relations:
         6*z6.1 = 0
[2]    Order 3          Length 1
      Abelian Group isomorphic to Z/3
      Defined on 1 generator in supergroup z6:
         $.1 = 2*z6.1
      Relations:
         3*$.1 = 0
[3]    Order 2          Length 1
      Abelian Group isomorphic to Z/2
      Defined on 1 generator in supergroup z6:
         $.1 = 3*z6.1
      Relations:
         2*$.1 = 0
[4]    Order 1          Length 1
      Abelian Group of order 1
true
Symmetric group G acting on a set of cardinality 3
Order = 6 = 2 * 3
Mapping from: GrpPerm: G to { 1 .. 6 }
[
[ 1, 2, 3, 4, 5, 6 ],
[ 2, 3, 1, 6, 4, 5 ],
[ 3, 1, 2, 5, 6, 4 ],
[ 4, 5, 6, 1, 2, 3 ],
[ 5, 6, 4, 3, 1, 2 ],
[ 6, 4, 5, 2, 3, 1 ]
]
Conjugacy classes of subgroups
------------------------------

[1]    Order 1          Length 1
      Permutation group acting on a set of cardinality 3
      Order = 1
[2]    Order 3          Length 1
      Permutation group acting on a set of cardinality 3
      Order = 3
         (1, 2, 3)
[3]    Order 6          Length 1
      Permutation group acting on a set of cardinality 3
      Order = 6 = 2 * 3
         (2, 3)
         (1, 2, 3)
[
Symmetric group G acting on a set of cardinality 3
Order = 6 = 2 * 3
      (1, 2, 3)
      (1, 2),
Permutation group acting on a set of cardinality 3
Order = 3
      (1, 2, 3),
Permutation group acting on a set of cardinality 3
Order = 1
]

万--本原表：

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格论：
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z1:=IntegerRing(40) ;
z6:=MultiplicativeGroup(z1);
z6;
# z6;
NumberOfGenerators(z6) ;
f := NumberingMap(z6);
f;
[ [ f(x*y) : y in z6 ] : x in z6 ];

A:=AutomorphismGroup(z6);
A;
Generators(z6);
11;
DerivedSeries(z6);
r1:=z6 !z6.1;
r1;
2*r1;
3*r1;
4*r1;
r11:=z6 !z6.2;
r11;
2*r11;
3*r11;
4*r11;
r111:=z6 !z6.3;
r111;
2*r111;
3*r111;
4*r111;

4444;
0;
r1;
r11;
r111;
2*r111;
3*r111;
r1+r11;
r1+r111;
r11+r111;
r1+2*r111;
r1+3*r111;
r11+2*r111;
r11+3*r111;
r1+r11+r111;
r1+r11+r111;
r1+r11+2*r111;
r1+r11+3*r111;

Abelian Group isomorphic to Z/2 + Z/2 + Z/4
Defined on 3 generators
Relations:
2*z6.1 = 0
2*z6.2 = 0
4*z6.3 = 0
16
3
Mapping from: GrpAb: z6 to { 1 .. 16 }
[
[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 ],
[ 2, 1, 4, 3, 6, 5, 8, 7, 10, 9, 12, 11, 14, 13, 16, 15 ],
[ 3, 4, 1, 2, 7, 8, 5, 6, 11, 12, 9, 10, 15, 16, 13, 14 ],
[ 4, 3, 2, 1, 8, 7, 6, 5, 12, 11, 10, 9, 16, 15, 14, 13 ],
[ 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1, 2, 3, 4 ],
[ 6, 5, 8, 7, 10, 9, 12, 11, 14, 13, 16, 15, 2, 1, 4, 3 ],
[ 7, 8, 5, 6, 11, 12, 9, 10, 15, 16, 13, 14, 3, 4, 1, 2 ],
[ 8, 7, 6, 5, 12, 11, 10, 9, 16, 15, 14, 13, 4, 3, 2, 1 ],
[ 9, 10, 11, 12, 13, 14, 15, 16, 1, 2, 3, 4, 5, 6, 7, 8 ],
[ 10, 9, 12, 11, 14, 13, 16, 15, 2, 1, 4, 3, 6, 5, 8, 7 ],
[ 11, 12, 9, 10, 15, 16, 13, 14, 3, 4, 1, 2, 7, 8, 5, 6 ],
[ 12, 11, 10, 9, 16, 15, 14, 13, 4, 3, 2, 1, 8, 7, 6, 5 ],
[ 13, 14, 15, 16, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ],
[ 14, 13, 16, 15, 2, 1, 4, 3, 6, 5, 8, 7, 10, 9, 12, 11 ],
[ 15, 16, 13, 14, 3, 4, 1, 2, 7, 8, 5, 6, 11, 12, 9, 10 ],
[ 16, 15, 14, 13, 4, 3, 2, 1, 8, 7, 6, 5, 12, 11, 10, 9 ]
]
A group of automorphisms of Abelian Group isomorphic to Z/2 + Z/2 + Z/4
Defined on 3 generators
Relations:
2*z6.1 = 0
2*z6.2 = 0
4*z6.3 = 0
Generators:
Automorphism of Abelian Group isomorphic to Z/2 + Z/2 + Z/4
Defined on 3 generators
Relations:
      2*z6.1 = 0
      2*z6.2 = 0
      4*z6.3 = 0 which maps:
      z6.1 |--> z6.2
      z6.2 |--> z6.1
      z6.3 |--> z6.3
Automorphism of Abelian Group isomorphic to Z/2 + Z/2 + Z/4
Defined on 3 generators
Relations:
      2*z6.1 = 0
      2*z6.2 = 0
      4*z6.3 = 0 which maps:
      z6.1 |--> z6.1 + z6.2
      z6.2 |--> z6.2
      z6.3 |--> z6.3
Automorphism of Abelian Group isomorphic to Z/2 + Z/2 + Z/4
Defined on 3 generators
Relations:
      2*z6.1 = 0
      2*z6.2 = 0
      4*z6.3 = 0 which maps:
      z6.1 |--> z6.1
      z6.2 |--> z6.2
      z6.3 |--> 3*z6.3
Automorphism of Abelian Group isomorphic to Z/2 + Z/2 + Z/4
Defined on 3 generators
Relations:
      2*z6.1 = 0
      2*z6.2 = 0
      4*z6.3 = 0 which maps:
      z6.1 |--> z6.1 + 2*z6.3
      z6.2 |--> z6.2
      z6.3 |--> z6.3
Automorphism of Abelian Group isomorphic to Z/2 + Z/2 + Z/4
Defined on 3 generators
Relations:
      2*z6.1 = 0
      2*z6.2 = 0
      4*z6.3 = 0 which maps:
      z6.1 |--> z6.1
      z6.2 |--> z6.2
      z6.3 |--> z6.1 + z6.3
{
z6.3,
z6.2,
z6.1
}
11
[
Abelian Group isomorphic to Z/2 + Z/2 + Z/4
Defined on 3 generators
Relations:
      2*z6.1 = 0
      2*z6.2 = 0
      4*z6.3 = 0,

Abelian Group of order 1
]
z6.1
0
z6.1
0
z6.2
0
z6.2
0
z6.3
2*z6.3
3*z6.3
0
4444
0
z6.1
z6.2
z6.3
2*z6.3
3*z6.3
z6.1 + z6.2
z6.1 + z6.3
z6.2 + z6.3
z6.1 + 2*z6.3
z6.1 + 3*z6.3
z6.2 + 2*z6.3
z6.2 + 3*z6.3
z6.1 + z6.2 + z6.3
z6.1 + z6.2 + z6.3
z6.1 + z6.2 + 2*z6.3
z6.1 + z6.2 + 3*z6.3

迷向群 isotropy group

迷向群[地曲m可gr以甲;“，o，on:一印担na] 作用在集合M上的作为变换群的已给群G的保持点x不动的元素组成的集合G:.这个集合实际上是G的子群，并且称为点x的迷向群(isotr0Py gro叩).下列术语在同一个意义下使用:平稳子群(sta- tionary sub脚uP)，稳定化子(stabili茂r)，G中心化子 (G一“nt琦五次r).如果M是Ha璐do盯空间，G是连续作用在M上的拓扑群，则G二是闭子群.更进一步，如果M和G是局部紧的，G有可数基，并可迁地作用在M上，那么存在从M到商空间G/H唯一的同胚，其中H是迷向群之一;所有的G:，x〔M，同构于H. 设M是光滑流形，G是光滑作用在M上的一个Lie群，则点x‘M的迷向群G:诱导了切空间 Tx(M)的线性变换的群;后者称作x处的线性迷向群(如伐汀isoti习pygro叩).在通过点x处的高阶的切空间上，可以得到相应的高阶的切丛的构造群中的迷向群的自然表示;它们称为高阶迷向群(瓦咖r刃川cr isot- ropy grouPs)(也见迷向表示(切txopy representation)).
......

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