徐振环编的《群论导引》在线阅读及视屏

cjsh

http://v.ku6.com/special/show_2409951/A7CPLYCPyefS-Qqj.html
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
http://read.chaoxing.com/ebook/detail.jhtml?id=10654667
群类论
http://s.chaoxing.com/ebook/list_20105020.html

cjsh

1
http://sourceforge.net/apps/trac/groupexplorer/wiki/The%20First%20Five%20Symmetric%20Groups
http://www.gap-system.org/Download/WinInst.html
再贴点常识：
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)).
    ......
http://www1.chkd.cnki.net/kns50/XSearch.aspx?KeyWord=%e8%bf%b7%e5%90%91%e7%be%a4

2
求凯莱表：
z1:=IntegerRing(40) ;
z6:=MultiplicativeGroup(z1);
z6;
# z6;
NumberOfGenerators(z6) ;
f := NumberingMap(z6);
f;
[ [ f(x*y) : y in z6 ] : x in z6 ];

3

万--本原表：
http://read.chaoxing.com/ebook/read_10831250.html
格论：
http://read.chaoxing.com/ebook/read_11177992.html
[ 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 ]

cjsh

[这个贴子最后由cjsh在 2011/07/04 04:57pm 第 3 次编辑]

4
26字母表乘群-------------密码学用
1 A B C D E F G H I J K L M N O P Q R S T U V W X Y
1 A B C D E F G H I J K L M N O P Q R S T U V W X Y
A L U K M X R S W Q T Y F O 1 V N C J I H G B E P D
B V 1 H R I M P C E Y T O F U L G W D X K N A Q S J
C Q T 1 X M S R V L U N I E K W Y A G F B J H O D P
D Y E F 1 B C H G J I L K N M P O R Q T S W X U V A
E X D G Q J N O F B A S P C W K H U 1 V L M Y R T I
F R S D V N T Q X K W M J B L U A Y H C E I G P 1 O
G U L E T C V 1 Y P M W B J S R I X O N D A F K Q H
H W K B S F X D A O N U E I T Q J V P M 1 Y C L R G
I S R P W Y U L M 1 V X G H Q T C N B A O F J D K E
J T Q O U A W K N D X V H G R S F M E Y P C I 1 L B
K C H A P O I J B F G 1 Q X Y E D L S R U T W V M N
L F G Y O P J I E C H D R V A B 1 K T Q W S U X N M
M D X R A U K W S T Q F Y 1 O N V J C H I E P G B L
N 1 V Q Y W L U T S R C A D P M X I F G J B O H E K
O M P J L G Y E I H C R D A V 1 B T K W Q X N S U F
P N O I K H A B J G F Q 1 Y X D E S L U R V M T W C
Q I J N E D G F O A B P S W C H K 1 U L V R T M Y X
R J I M B 1 H C P Y E O T U F G L D W K X Q S N A V
S G F X H K B A D N O E U T I J Q P V 1 M L R Y C W
T H C V G L E Y 1 M P B W S J I R O X D N K Q A F U
U B A W J Q O N K X D H V R G F S E M P Y 1 L C I T
V O N T F S D X Q W K J M L B A U H Y E C P 1 I G R
W E Y U I R P M L V 1 G X Q H C T B N O A D K F J S
X P M S C T 1 V R U L I N K E Y W G A B F O D J H Q
Y K W L N V Q T U R S A C P D X M F I J G H E B O 1
$
1
b^2
ab^5
ab^6
a
b^5
b^6
ab
b
ab^10
b^10
ab^4
b^4
ab^11
b^11
ab^9
b^9
ab^8
b^8
ab^12
b^12
ab^3
ab^7
b^3
b^7
ab^2
Dihedral(13)
Relations:
    a^2 = 1,
    b^13 = 1,
    b^a = b^12



5
集合运算表：
Operator Usage Meaning
# #S the number of elements in S
in x in S true if the element x is in S, else false
notin x notin S true if the element x is not in S, else false
cat S1 cat S2 the concatenation of sequences S1 and S2
join S1 join S2 the union of sets S1 and S2
meet S1 meet S2 the intersection of sets S1 and S2
diff S1 diff S2 the set of elements in set S1 but not in set S2
sdiff S1 sdiff S2 the symmetric difference of sets S1 and S2
subset S1 subset S2 true if S1 is a subset of S2, else false

关系表：
not not a true if a is false, else false
and a and b true if both a and b are true, else false
or a or b true if either a or b is true, else false
xor a xor b true if exactly one of a and b is true, else false
eq x eq y true if x is equal to y , else false
ne x ne y true if x is not equal to y , else false
lt x lt y true if x is less than y , else false
le x le y true if x is less than or equal to y , else false
gt x gt y true if x is greater than y , else false
ge x ge y true if x is greater than or equal to y , else false

6


小群表：
Size Construction Notes
1 SymmetricGroup(1) Trivial
2 SymmetricGroup(2) Also CyclicPermutationGroup(2)
3 CyclicPermutationGroup(3) Prime order
4 CyclicPermutationGroup(4) Cyclic
4 KleinFourGroup() Abelian, non-cyclic
5 CyclicPermutationGroup(5) Prime order
6 CyclicPermutationGroup(6) Cyclic
6 SymmetricGroup(3) Non-abelian, also DihedralGroup(3)
7 CyclicPermutationGroup(7) Prime order
8 CyclicPermutationGroup(8) Cyclic
8 D1=CyclicPermutationGroup(4)
D2=CyclicPermutationGroup(2)
G=direct product permgroups([D1,D2])
Abelian, non-cyclic
8 D1=CyclicPermutationGroup(2)
D2=CyclicPermutationGroup(2)
D3=CyclicPermutationGroup(2)
G=direct product permgroups([D1,D2,D3])
Abelian, non-cyclic
8 DihedralGroup(4) Non-abelian
8 PermutationGroup(["(1,2,5,6)(3,4,7,8)",
"(1,3,5,7)(2,8,6,4)" ])
Quaternions
The two generators are I and J
9 CyclicPermutationGroup(9) Cyclic
9 D1=CyclicPermutationGroup(3)
D2=CyclicPermutationGroup(3)
G=direct product permgroups([D1,D2])
Abelian, non-cyclic
10 CyclicPermutationGroup(10) Cyclic
10 DihedralGroup(5) Non-abelian
11 CyclicPermutationGroup(11) Prime order
12 CyclicPermutationGroup(12) Cyclic
12 D1=CyclicPermutationGroup(6)
D2=CyclicPermutationGroup(2)
G=direct product permgroups([D1,D2])
Abelian, non-cyclic
12 DihedralGroup(6) Non-abelian
12 AlternatingGroup(4) Non-abelian, symmetries of tetrahedron
12 PermutationGroup(["(1,2,3)(4,6)(5,7)",
"(1,2)(4,5,6,7)"])
Non-abelian
Semi-direct product Z3 o Z4
13 CyclicPermutationGroup(13) Prime order
14 CyclicPermutationGroup(14) Cyclic
14 DihedralGroup(7) Non-abelian
15 CyclicPermutationGroup(15) Cyclic

cjsh

小群表里15阶只有一个循环群，没Z3+Z5 12阶5个里有个怪点的：非交换半直积Z3+Z4 12 PermutationGroup(["(1,2,3)(4,6)(5,7)", "(1,2)(4,5,6,7)"]) Non-abelian Semi-direct product Z3 o Z4 G:=PermutationGroup<12|(1,2,3)(4,6)(5,7),(1,2)(4,5,6,7)>; G; f := NumberingMap(G); > [ [ f(x*y) : y in G ] : x in G ]; Permutation group G acting on a set of cardinality 12 (1, 2, 3)(4, 6)(5, 7) (1, 2)(4, 5, 6, 7) [ [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ], [ 2, 12, 1, 6, 4, 8, 9, 7, 5, 11, 3, 10 ], [ 3, 1, 11, 5, 9, 4, 8, 6, 7, 12, 10, 2 ], [ 4, 5, 6, 10, 11, 12, 1, 2, 3, 7, 8, 9 ], [ 5, 9, 4, 12, 10, 2, 3, 1, 11, 8, 6, 7 ], [ 6, 4, 8, 11, 3, 10, 2, 12, 1, 9, 7, 5 ], [ 7, 8, 9, 1, 2, 3, 10, 11, 12, 4, 5, 6 ], [ 8, 6, 7, 3, 1, 11, 12, 10, 2, 5, 9, 4 ], [ 9, 7, 5, 2, 12, 1, 11, 3, 10, 6, 4, 8 ], [ 10, 11, 12, 7, 8, 9, 4, 5, 6, 1, 2, 3 ], [ 11, 3, 10, 9, 7, 5, 6, 4, 8, 2, 12, 1 ], [ 12, 10, 2, 8, 6, 7, 5, 9, 4, 3, 1, 11 ] ] G:=PermutationGroup<12|(1,2,3)(4,6)(5,7),(1,2)(4,5,6,7)>; G; f := NumberingMap(G); > [ [ f(x*y) : y in G ] : x in G ]; Z3:=IntegerRing(3) ; A3:=AdditiveGroup(Z3) ; Z4:=IntegerRing(4) ; A4:=AdditiveGroup(Z4) ; SemiDirectProduct(A3, A4); Permutation group G acting on a set of cardinality 12 (1, 2, 3)(4, 6)(5, 7) (1, 2)(4, 5, 6, 7) [ [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ], [ 2, 12, 1, 6, 4, 8, 9, 7, 5, 11, 3, 10 ], [ 3, 1, 11, 5, 9, 4, 8, 6, 7, 12, 10, 2 ], [ 4, 5, 6, 10, 11, 12, 1, 2, 3, 7, 8, 9 ], [ 5, 9, 4, 12, 10, 2, 3, 1, 11, 8, 6, 7 ], [ 6, 4, 8, 11, 3, 10, 2, 12, 1, 9, 7, 5 ], [ 7, 8, 9, 1, 2, 3, 10, 11, 12, 4, 5, 6 ], [ 8, 6, 7, 3, 1, 11, 12, 10, 2, 5, 9, 4 ], [ 9, 7, 5, 2, 12, 1, 11, 3, 10, 6, 4, 8 ], [ 10, 11, 12, 7, 8, 9, 4, 5, 6, 1, 2, 3 ], [ 11, 3, 10, 9, 7, 5, 6, 4, 8, 2, 12, 1 ], [ 12, 10, 2, 8, 6, 7, 5, 9, 4, 3, 1, 11 ] ] >> SemiDirectProduct(A3, A4);

cjsh

[这个贴子最后由cjsh在 2011/07/04 04:59pm 第 1 次编辑]

7
环论符号：

Symbol         Description               Category
--------------------------------------------------
Z              ring of integers          RngInt
Z/mZ           ring of residue classes   RngRes
R[x]           univariate poly. ring     RngUPol
F[x]/f(x)      univ. poly. factor ring   RngUPolRes
R[x_1,...,x_m] multivariate poly. ring   RngMPol
R[[x]]         power series ring         RngSer
O              order in a number field   RngOrd
\Z_p           p-adic ring               RngPad
R_m            local ring                RngLoc
V              valuation ring            RngVal
--------------------------------------------------
Q              rational field            FldRat
F_q            finite field              FldFin
F(x_1,...,x_m) rational function field   FldFun
F((x))         field of Laurent series   FldPow
Q(sqrt(D))     quadratic number field    FldQuad
Q(zeta_n)      cyclotomic number field   FldCyc
Q(alpha)       number field              FldNum
Q_p            p-adic field              FldPad
Q_p(alpha)     local field               FldLoc
R              real field                FldRe
C              complex field             FldCom