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lilianjie 雪币： 433 活跃值： (45) 能力值： ( LV4，RANK：50 ) 在线值：发帖 141 回帖 549 粉丝 1 关注私信	lilianjie 1 2 楼 S4第一行调不了。。。上传的附件： ts.jpg （4.19kb，159次下载） conjugate.png （13.41kb，157次下载） normalSub.png （19.52kb，157次下载） 2011-1-6 15:38 0
lilianjie 雪币： 433 活跃值： (45) 能力值： ( LV4，RANK：50 ) 在线值：发帖 141 回帖 549 粉丝 1 关注私信	lilianjie 1 3 楼 GOOGLE.COM 老外老师就是GNU，多具象：上传的附件： Cayley_S_4.gif （35.89kb，150次下载） figure0327379.jpg （32.19kb，151次下载） 2011-1-6 16:10 0
lilianjie 雪币： 433 活跃值： (45) 能力值： ( LV4，RANK：50 ) 在线值：发帖 141 回帖 549 粉丝 1 关注私信	lilianjie 1 4 楼运动的..... 上传的附件： cd-s4-transpo.jpg （14.58kb，14次下载） 2011-1-6 17:02 0
lilianjie 雪币： 433 活跃值： (45) 能力值： ( LV4，RANK：50 ) 在线值：发帖 141 回帖 549 粉丝 1 关注私信	lilianjie 1 5 楼 http://sourceforge.net/apps/trac/groupexplorer/wiki/The%20First%20Five%20Symmetric%20Groups http://www.gap-system.org/Download/WinInst.html 2011-1-6 17:03 0
刘觐肇雪币： 119 活跃值： (10) 能力值： ( LV2，RANK：10 ) 在线值：发帖 11 回帖 149 粉丝 0 关注私信	刘觐肇 6 楼很闪耀 2011-1-6 17:22 0
lilianjie 雪币： 433 活跃值： (45) 能力值： ( LV4，RANK：50 ) 在线值：发帖 141 回帖 549 粉丝 1 关注私信	lilianjie 1 7 楼两月了。。还在转呢 26字母表乘群 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 A:=IntegerRing(13); R:=ResidueClassRing(137); R; U:=UnitGroup(R); U; U1:=UnitGroup(A); U; a:=A!2999; Order(2999) ; Order(a) ; #U; aa; EulerPhi(2008); EulerPhi(130); PrimitiveElement(R); PrimitiveElement(A); PrimitiveRoot(R); PrimitiveRoot(A); 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 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 There is an extremely important class of operators called reduction operators. These have the form &op and act on a set or sequence to reduce it to a single element by repeated application of the binary operator op. Not all binary operators have a corresponding reduction operator; the valid reduction operators are as follows. &+ &∗ &and &or &meet &join &cat If S contains the elements α1, . . . , αn in that order3 then &op S = (. . . ((α1 op α2) op α3) . . .) op αn. Note that most of these operators are commutative (at least under common circumstances). The exception is &cat , which concatenates the elements of the set/sequence. 3For sets, the order is the internal iteration order, which corresponds to the order used when printing. 344 Geoff Bailey > &+[ k 2 : k in [1. . 24] ]; 4900 > F 11 := GF(11); > _<t> := PolynomialRing(F 11); > &∗{ t − a : a in F 11 \| not IsSquare(a) }; t^5 + 1 > P<a,b,c> := PolynomialRing(Integers(), 3); > sympols := [ &+[ &∗S : S in Subsets({a,b,c }, k ) ] : k in [0. . 3] ]; > sympols ; [ 1, a + b + c, ab + ac + bc, abc ] > &and [ IsSymmetric(pol) : pol in sympols ]; true > &cat [ “Hello,”, “ ”, “world!” ]; Hello, world! > &meet [ { n : n in [2. . 104] \| Modexp(x ,n,n) eq x } : x in {2, 3, 5} ] > diff { n : n in [2. . 104] \| IsPrime(n) }; { 561, 1105, 1729, 2465, 2821, 6601, 8911 } When a set or sequence is empty then it may not be obvious what the result of applying a reduction operator to it should be. The guiding principle is that the result should be consistent, in the sense that (&op S) op x should produce x when S is empty. Thus &+ produces 0 and &∗ produces 1, &and produces true and &or produces false, and &join and &cat produce empty objects of the appropriate types. It is not permissible to call &meet on an empty structure. In the case of &+ and &∗, if they are called on an empty set or sequence then the universe must have already been set. This is so that Magma can determine in which structure the result should lie. 上传的附件： The irreducible subgroups of GL(V).zip （1.88kb，5次下载） 2011-4-2 11:31 0
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lilianjie

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lilianjie 1: 2 楼

S4第一行调不了。。。

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2011-1-6 15:38

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lilianjie

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lilianjie 1: 3 楼

GOOGLE.COM

老外老师就是GNU，多具象：

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2011-1-6 16:10

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lilianjie

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lilianjie 1: 4 楼

运动的.....

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2011-1-6 17:02

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lilianjie

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lilianjie 1: 5 楼

http://sourceforge.net/apps/trac/groupexplorer/wiki/The%20First%20Five%20Symmetric%20Groups

http://www.gap-system.org/Download/WinInst.html

2011-1-6 17:03

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刘觐肇

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刘觐肇: 6 楼

很闪耀

2011-1-6 17:22

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lilianjie

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lilianjie 1: 7 楼

两月了。。还在转呢

26字母表乘群

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

A:=IntegerRing(13);
R:=ResidueClassRing(137);
R;
U:=UnitGroup(R);
U;
U1:=UnitGroup(A);
U;
a:=A!2999;
Order(2999) ;
Order(a) ;
#U;
a*a;
EulerPhi(2008);
EulerPhi(130);
PrimitiveElement(R);
PrimitiveElement(A);
PrimitiveRoot(R);
PrimitiveRoot(A);

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

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

There is an extremely important class of operators called reduction operators.
These have the form &op and act on a set or sequence to reduce it to a
single element by repeated application of the binary operator op. Not all
binary operators have a corresponding reduction operator; the valid reduction
operators are as follows.
&+ &∗ &and &or &meet &join &cat
If S contains the elements α1, . . . , αn in that order3 then
&op S = (. . . ((α1 op α2) op α3) . . .) op αn.
Note that most of these operators are commutative (at least under common
circumstances). The exception is &cat , which concatenates the elements of the
set/sequence.
3For sets, the order is the internal iteration order, which corresponds to the order
used when printing.
344 Geoff Bailey
> &+[ k 2 : k in [1. . 24] ];
4900
> F 11 := GF(11);
> _<t> := PolynomialRing(F 11);
> &∗{ t − a : a in F 11 | not IsSquare(a) };
t^5 + 1
> P<a,b,c> := PolynomialRing(Integers(), 3);
> sympols := [ &+[ &∗S : S in Subsets({a,b,c }, k ) ] : k in [0. . 3] ];
> sympols ;
[
1,
a + b + c,
a*b + a*c + b*c,
a*b*c
]
> &and [ IsSymmetric(pol) : pol in sympols ];
true
> &cat [ “Hello,”, “ ”, “world!” ];
Hello, world!
> &meet [ { n : n in [2. . 104] | Modexp(x ,n,n) eq x } : x in {2, 3, 5} ]
> diff { n : n in [2. . 104] | IsPrime(n) };
{ 561, 1105, 1729, 2465, 2821, 6601, 8911 }
When a set or sequence is empty then it may not be obvious what the result
of applying a reduction operator to it should be. The guiding principle is that
the result should be consistent, in the sense that (&op S) op x should produce
x when S is empty. Thus &+ produces 0 and &∗ produces 1, &and produces
true and &or produces false, and &join and &cat produce empty objects of the
appropriate types. It is not permissible to call &meet on an empty structure.
In the case of &+ and &∗, if they are called on an empty set or sequence
then the universe must have already been set. This is so that Magma can
determine in which structure the result should lie.

上传的附件：

The irreducible subgroups of GL(V).zip （1.88kb，5次下载）

2011-4-2 11:31

0