群¶
置换群¶
置换群是某一对称群 \(S_n\) 的子群。Sage 有一个 Python 类 PermutationGroup，
因此你可以直接使用此类群:
sage: G = PermutationGroup(['(1,2,3)(4,5)'])
sage: G
Permutation Group with generators [(1,2,3)(4,5)]
sage: g = G.gens()[0]; g
(1,2,3)(4,5)
sage: g*g
(1,3,2)
sage: G = PermutationGroup(['(1,2,3)'])
sage: g = G.gens()[0]; g
(1,2,3)
sage: g.order()
3
>>> from sage.all import *
>>> G = PermutationGroup(['(1,2,3)(4,5)'])
>>> G
Permutation Group with generators [(1,2,3)(4,5)]
>>> g = G.gens()[Integer(0)]; g
(1,2,3)(4,5)
>>> g*g
(1,3,2)
>>> G = PermutationGroup(['(1,2,3)'])
>>> g = G.gens()[Integer(0)]; g
(1,2,3)
>>> g.order()
3
对于魔方群（\(S_{48}\) 的置换子群，其中魔方的非中心面以某种固定方式标记为 \(1,2,...,48\)）， 你可以按如下方式使用 GAP-Sage 接口。
sage: cube = "cubegp := Group(
( 1, 3, 8, 6)( 2, 5, 7, 4)( 9,33,25,17)(10,34,26,18)(11,35,27,19),
( 9,11,16,14)(10,13,15,12)( 1,17,41,40)( 4,20,44,37)( 6,22,46,35),
(17,19,24,22)(18,21,23,20)( 6,25,43,16)( 7,28,42,13)( 8,30,41,11),
(25,27,32,30)(26,29,31,28)( 3,38,43,19)( 5,36,45,21)( 8,33,48,24),
(33,35,40,38)(34,37,39,36)( 3, 9,46,32)( 2,12,47,29)( 1,14,48,27),
(41,43,48,46)(42,45,47,44)(14,22,30,38)(15,23,31,39)(16,24,32,40) )"
sage: gap(cube)
'permutation group with 6 generators'
sage: gap("Size(cubegp)")
43252003274489856000'
>>> from sage.all import *
>>> cube = "cubegp := Group(
( 1, 3, 8, 6)( 2, 5, 7, 4)( 9,33,25,17)(10,34,26,18)(11,35,27,19),
( 9,11,16,14)(10,13,15,12)( 1,17,41,40)( 4,20,44,37)( 6,22,46,35),
(17,19,24,22)(18,21,23,20)( 6,25,43,16)( 7,28,42,13)( 8,30,41,11),
(25,27,32,30)(26,29,31,28)( 3,38,43,19)( 5,36,45,21)( 8,33,48,24),
(33,35,40,38)(34,37,39,36)( 3, 9,46,32)( 2,12,47,29)( 1,14,48,27),
(41,43,48,46)(42,45,47,44)(14,22,30,38)(15,23,31,39)(16,24,32,40) )"
>>> gap(cube)
'permutation group with 6 generators'
>>> gap("Size(cubegp)")
43252003274489856000'
你还可以选择另一种方式来实现：
创建一个包含以下内容的文件
cubegroup.py:cube = "cubegp := Group( ( 1, 3, 8, 6)( 2, 5, 7, 4)( 9,33,25,17)(10,34,26,18)(11,35,27,19), ( 9,11,16,14)(10,13,15,12)( 1,17,41,40)( 4,20,44,37)( 6,22,46,35), (17,19,24,22)(18,21,23,20)( 6,25,43,16)( 7,28,42,13)( 8,30,41,11), (25,27,32,30)(26,29,31,28)( 3,38,43,19)( 5,36,45,21)( 8,33,48,24), (33,35,40,38)(34,37,39,36)( 3, 9,46,32)( 2,12,47,29)( 1,14,48,27), (41,43,48,46)(42,45,47,44)(14,22,30,38)(15,23,31,39)(16,24,32,40) )"
然后将该文件放置在 Sage 目录的
$SAGE_ROOT/local/lib/python2.4/site-packages/sage子目录中。最后，读取（即import）该文件到 Sage 中：sage: import sage.cubegroup sage: sage.cubegroup.cube 'cubegp := Group(( 1, 3, 8, 6)( 2, 5, 7, 4)( 9,33,25,17)(10,34,26,18) (11,35,27,19),( 9,11,16,14)(10,13,15,12)( 1,17,41,40)( 4,20,44,37) ( 6,22,46,35),(17,19,24,22)(18,21,23,20)( 6,25,43,16)( 7,28,42,13) ( 8,30,41,11),(25,27,32,30)(26,29,31,28)( 3,38,43,19)( 5,36,45,21) ( 8,33,48,24),(33,35,40,38)(34,37,39,36)( 3, 9,46,32)( 2,12,47,29) ( 1,14,48,27),(41,43,48,46)(42,45,47,44)(14,22,30,38)(15,23,31,39) (16,24,32,40) )' sage: gap(sage.cubegroup.cube) 'permutation group with 6 generators' sage: gap("Size(cubegp)") '43252003274489856000'
>>> from sage.all import * >>> import sage.cubegroup >>> sage.cubegroup.cube 'cubegp := Group(( 1, 3, 8, 6)( 2, 5, 7, 4)( 9,33,25,17)(10,34,26,18) (11,35,27,19),( 9,11,16,14)(10,13,15,12)( 1,17,41,40)( 4,20,44,37) ( 6,22,46,35),(17,19,24,22)(18,21,23,20)( 6,25,43,16)( 7,28,42,13) ( 8,30,41,11),(25,27,32,30)(26,29,31,28)( 3,38,43,19)( 5,36,45,21) ( 8,33,48,24),(33,35,40,38)(34,37,39,36)( 3, 9,46,32)( 2,12,47,29) ( 1,14,48,27),(41,43,48,46)(42,45,47,44)(14,22,30,38)(15,23,31,39) (16,24,32,40) )' >>> gap(sage.cubegroup.cube) 'permutation group with 6 generators' >>> gap("Size(cubegp)") '43252003274489856000'
（在 Sage 输出中，会使用换行符来代替上面的回车符。）
使用
CubeGroup类:sage: rubik = CubeGroup() sage: rubik The Rubik's cube group with generators R,L,F,B,U,D in SymmetricGroup(48). sage: rubik.order() 43252003274489856000
>>> from sage.all import * >>> rubik = CubeGroup() >>> rubik The Rubik's cube group with generators R,L,F,B,U,D in SymmetricGroup(48). >>> rubik.order() 43252003274489856000
(1) Sage 实现了经典群（如 \(GU(3,\GF{5})\)） 和具有用户定义生成器的有限域上的矩阵群。
(2) Sage 还实现了有限和无限 （但有限生成）阿贝尔群。
共轭类¶
你可以“原生地”计算有限群的共轭类:
sage: G = PermutationGroup(['(1,2,3)', '(1,2)(3,4)', '(1,7)'])
sage: CG = G.conjugacy_classes_representatives()
sage: gamma = CG[2]
sage: CG; gamma
[(), (4,7), (3,4,7), (2,3)(4,7), (2,3,4,7), (1,2)(3,4,7), (1,2,3,4,7)]
(3,4,7)
>>> from sage.all import *
>>> G = PermutationGroup(['(1,2,3)', '(1,2)(3,4)', '(1,7)'])
>>> CG = G.conjugacy_classes_representatives()
>>> gamma = CG[Integer(2)]
>>> CG; gamma
[(), (4,7), (3,4,7), (2,3)(4,7), (2,3,4,7), (1,2)(3,4,7), (1,2,3,4,7)]
(3,4,7)
你可以使用 Sage-GAP 接口完成这一任务:
sage: libgap.eval("G := Group((1,2)(3,4),(1,2,3))")
Group([ (1,2)(3,4), (1,2,3) ])
sage: libgap.eval("CG := ConjugacyClasses(G)")
[ ()^G, (2,3,4)^G, (2,4,3)^G, (1,2)(3,4)^G ]
sage: libgap.eval("gamma := CG[3]")
(2,4,3)^G
sage: libgap.eval("g := Representative(gamma)")
(2,4,3)
>>> from sage.all import *
>>> libgap.eval("G := Group((1,2)(3,4),(1,2,3))")
Group([ (1,2)(3,4), (1,2,3) ])
>>> libgap.eval("CG := ConjugacyClasses(G)")
[ ()^G, (2,3,4)^G, (2,4,3)^G, (1,2)(3,4)^G ]
>>> libgap.eval("gamma := CG[3]")
(2,4,3)^G
>>> libgap.eval("g := Representative(gamma)")
(2,4,3)
或者，这里有另一种（更符合 Python 风格的）方法来进行该计算:
sage: G = libgap.eval("Group([(1,2,3), (1,2)(3,4), (1,7)])")
sage: CG = G.ConjugacyClasses()
sage: gamma = CG[2]
sage: g = gamma.Representative()
sage: CG; gamma; g
[ ()^G, (4,7)^G, (3,4,7)^G, (2,3)(4,7)^G, (2,3,4,7)^G, (1,2)(3,4,7)^G, (1,2,3,4,7)^G ]
(3,4,7)^G
(3,4,7)
>>> from sage.all import *
>>> G = libgap.eval("Group([(1,2,3), (1,2)(3,4), (1,7)])")
>>> CG = G.ConjugacyClasses()
>>> gamma = CG[Integer(2)]
>>> g = gamma.Representative()
>>> CG; gamma; g
[ ()^G, (4,7)^G, (3,4,7)^G, (2,3)(4,7)^G, (2,3,4,7)^G, (1,2)(3,4,7)^G, (1,2,3,4,7)^G ]
(3,4,7)^G
(3,4,7)
正规子群¶
如果想要找到置换群 \(G\) （从共轭角度）的所有正规子群，可以使用 Sage 的 GAP 接口:
sage: G = AlternatingGroup( 5 )
sage: libgap(G).NormalSubgroups()
[ Alt( [ 1 .. 5 ] ), Group(()) ]
>>> from sage.all import *
>>> G = AlternatingGroup( Integer(5) )
>>> libgap(G).NormalSubgroups()
[ Alt( [ 1 .. 5 ] ), Group(()) ]
或者
sage: G = libgap.AlternatingGroup( 5 )
sage: G.NormalSubgroups()
[ Alt( [ 1 .. 5 ] ), Group(()) ]
>>> from sage.all import *
>>> G = libgap.AlternatingGroup( Integer(5) )
>>> G.NormalSubgroups()
[ Alt( [ 1 .. 5 ] ), Group(()) ]
这里有另一种更直接使用 GAP 的方法:
sage: libgap.eval("G := AlternatingGroup( 5 )")
Alt( [ 1 .. 5 ] )
sage: libgap.eval("normal := NormalSubgroups( G )")
[ Alt( [ 1 .. 5 ] ), Group(()) ]
sage: G = libgap.eval("DihedralGroup( 10 )")
sage: G.NormalSubgroups().SortedList()
[ Group([ ]), Group([ f2 ]), <pc group of size 10 with 2 generators> ]
sage: libgap.eval("G := SymmetricGroup( 4 )")
Sym( [ 1 .. 4 ] )
sage: libgap.eval("normal := NormalSubgroups( G );")
[ Sym( [ 1 .. 4 ] ), Alt( [ 1 .. 4 ] ), Group([ (1,4)(2,3), ... ]),
Group(()) ]
>>> from sage.all import *
>>> libgap.eval("G := AlternatingGroup( 5 )")
Alt( [ 1 .. 5 ] )
>>> libgap.eval("normal := NormalSubgroups( G )")
[ Alt( [ 1 .. 5 ] ), Group(()) ]
>>> G = libgap.eval("DihedralGroup( 10 )")
>>> G.NormalSubgroups().SortedList()
[ Group([ ]), Group([ f2 ]), <pc group of size 10 with 2 generators> ]
>>> libgap.eval("G := SymmetricGroup( 4 )")
Sym( [ 1 .. 4 ] )
>>> libgap.eval("normal := NormalSubgroups( G );")
[ Sym( [ 1 .. 4 ] ), Alt( [ 1 .. 4 ] ), Group([ (1,4)(2,3), ... ]),
Group(()) ]
中心¶
如何在 Sage 中计算群的中心？
虽然 Sage 调用 GAP 来计算群的中心，
但 center 是“封装”过的方法（即 Sage 有一个类 PermutationGroup 关联 "center" 方法），
因此用户不需要使用 libgap 命令。这里有一个例子:
sage: G = PermutationGroup(['(1,2,3)(4,5)', '(3,4)'])
sage: G.center()
Subgroup generated by [()] of (Permutation Group with generators [(3,4), (1,2,3)(4,5)])
>>> from sage.all import *
>>> G = PermutationGroup(['(1,2,3)(4,5)', '(3,4)'])
>>> G.center()
Subgroup generated by [()] of (Permutation Group with generators [(3,4), (1,2,3)(4,5)])
类似的语法也适用于矩阵群:
sage: G = SL(2, GF(5) )
sage: G.center()
Subgroup with 1 generators (
[4 0]
[0 4]
) of Special Linear Group of degree 2 over Finite Field of size 5
sage: G = PSL(2, 5 )
sage: G.center()
Subgroup generated by [()] of (The projective special linear group of degree 2 over Finite Field of size 5)
>>> from sage.all import *
>>> G = SL(Integer(2), GF(Integer(5)) )
>>> G.center()
Subgroup with 1 generators (
[4 0]
[0 4]
) of Special Linear Group of degree 2 over Finite Field of size 5
>>> G = PSL(Integer(2), Integer(5) )
>>> G.center()
Subgroup generated by [()] of (The projective special linear group of degree 2 over Finite Field of size 5)
备注
在 GAP 中 center 有两种拼写方式，但在 Sage 中不行。
群 id 数据库¶
函数 group_id 使用了 E. A. O'Brien、B. Eick 和 H. U. Besche 的小群库，它是 GAP 的一部分。
sage: G = PermutationGroup(['(1,2,3)(4,5)', '(3,4)'])
sage: G.order()
120
sage: G.group_id()
[120, 34]
>>> from sage.all import *
>>> G = PermutationGroup(['(1,2,3)(4,5)', '(3,4)'])
>>> G.order()
120
>>> G.group_id()
[120, 34]
另一个使用小型群数据库的例子：group_id
sage: gap_console()
┌───────┐ GAP 4.10.0 of 01-Nov-2018
│ GAP │ https://www.gap-system.org
└───────┘ Architecture: x86_64-pc-linux-gnu-default64
Configuration: gmp 6.0.0, readline
Loading the library and packages ...
Packages: GAPDoc 1.6.2, PrimGrp 3.3.2, SmallGrp 1.3, TransGrp 2.0.4
Try '??help' for help. See also '?copyright', '?cite' and '?authors'
gap> G:=Group((4,6,5)(7,8,9),(1,7,2,4,6,9,5,3));
Group([ (4,6,5)(7,8,9), (1,7,2,4,6,9,5,3) ])
gap> StructureDescription(G);
"(C3 x C3) : GL(2,3)"
>>> from sage.all import *
>>> gap_console()
┌───────┐ GAP 4.10.0 of 01-Nov-2018
│ GAP │ https://www.gap-system.org
└───────┘ Architecture: x86_64-pc-linux-gnu-default64
Configuration: gmp 6.0.0, readline
Loading the library and packages ...
Packages: GAPDoc 1.6.2, PrimGrp 3.3.2, SmallGrp 1.3, TransGrp 2.0.4
Try '??help' for help. See also '?copyright', '?cite' and '?authors'
gap> G:=Group((4,6,5)(7,8,9),(1,7,2,4,6,9,5,3));
Group([ (4,6,5)(7,8,9), (1,7,2,4,6,9,5,3) ])
gap> StructureDescription(G);
"(C3 x C3) : GL(2,3)"
小于 32 阶的群的构建指令¶
作者：
Davis Shurbert
每个小于 32 阶的群都在 Sage 中实现为置换群。这些群的构建都非常简单。 我们首先展示如何构建直积和半直积，然后给出构建这些小群所需的命令。
设 G1, G2, ..., Gn 是已经在 Sage 中初始化的置换群。
可以使用以下命令取它们的直积
（当然，这里省略号只是作为符号使用，实际上必须显式输入所求乘积中的每个因子）。
sage: G = direct_product_permgroups([G1, G2, ..., Gn])
>>> from sage.all import *
>>> G = direct_product_permgroups([G1, G2, Ellipsis, Gn])
半直积运算可以被视为直积运算的推广。给定两个群 \(H\) 和 \(K\)，它们的半直积 \(H \ltimes_{\phi} K\) （其中 \(\phi : H \rightarrow Aut(K)\) 是一个同态）是一个群，其基础集合是 \(H\) 和 \(K\) 的笛卡尔积， 但具有以下运算：
输出不是运算定义中明确描述的群，而是一个同构的置换群。
在下面的例程中，假设 H 和 K 已经在 Sage 中定义且初始化。
此外，phi 是一个包含两个子列表的列表，通过给出 H 的生成器集合的像来定义底层同态。
对于下表中的每个半直积群，我们将展示如何构建 phi，然后假设你已经阅读此段落并理解如何从那里开始。
sage: G = H.semidirect_product(K, phi)
>>> from sage.all import *
>>> G = H.semidirect_product(K, phi)
为了避免不必要的重复，我们现在将给出创建 \(n\) 阶循环群 \(C_n\) 的命令和 \(n\) 个字母的二面体群 \(D_n\) 的命令。 我们还会为每个命令展示一个例子以确保读者理解这些命令，然后不再重复。
sage: G = CyclicPermutationGroup(n)
sage: G = DihedralGroup(n)
>>> from sage.all import *
>>> G = CyclicPermutationGroup(n)
>>> G = DihedralGroup(n)
请注意，直积运算中将使用指数表示法。例如 \({C_2}^2 = C_2 \times C_2\)。 该表格是在 AD Thomas 和 GV Wood 的 Group Tables (1980, Shiva Publishing) 的帮助下制作的。
阶 |
群描述 |
命令 |
GAP ID |
|---|---|---|---|
1 |
平凡群 |
sage: G = SymmetricGroup(1)
>>> from sage.all import *
>>> G = SymmetricGroup(Integer(1))
|
[1,1] |
2 |
\(C_2\) |
sage: G = SymmetricGroup(2)
>>> from sage.all import *
>>> G = SymmetricGroup(Integer(2))
|
[2,1] |
3 |
\(C_3\) |
sage: G = CyclicPermutationGroup(3)
>>> from sage.all import *
>>> G = CyclicPermutationGroup(Integer(3))
|
[3,1] |
4 |
\(C_4\) |
[4,1] |
|
4 |
\(C_2 \times C_2\) |
sage: G = KleinFourGroup()
>>> from sage.all import *
>>> G = KleinFourGroup()
|
[4,2] |
5 |
\(C_5\) |
[5,1] |
|
6 |
\(C_6\) |
[6,2] |
|
6 |
\(S_3\) （三字母对称群） |
sage: G = SymmetricGroup(3)
>>> from sage.all import *
>>> G = SymmetricGroup(Integer(3))
|
[6,1] |
7 |
\(C_7\) |
[7,1] |
|
8 |
\(C_8\) |
[8,1] |
|
8 |
\(C_4 \times C_2\) |
[8,2] |
|
8 |
\(C_2\times C_2\times C_2\) |
[8,5] |
|
8 |
\(D_4\) |
sage: G = DihedralGroup(4)
>>> from sage.all import *
>>> G = DihedralGroup(Integer(4))
|
[8,3] |
8 |
四元群 (Q) |
sage: G = QuaternionGroup()
>>> from sage.all import *
>>> G = QuaternionGroup()
|
[8,4] |
9 |
\(C_9\) |
[9,1] |
|
9 |
\(C_3 \times C_3\) |
[9,2] |
|
10 |
\(C_{10}\) |
[10,2] |
|
10 |
\(D_5\) |
[10,1] |
|
11 |
\(C_{11}\) |
[11,1] |
|
12 |
\(C_{12}\) |
[12,2] |
|
12 |
\(C_6 \times C_2\) |
[12,5] |
|
12 |
\(D_6\) |
[12,4] |
|
12 |
\(A_4\) （四字母交错群） |
sage: G = AlternatingGroup(4)
>>> from sage.all import *
>>> G = AlternatingGroup(Integer(4))
|
[12,3] |
12 |
\(Q_6\) （12 阶双环群） |
sage: G = DiCyclicGroup(3)
>>> from sage.all import *
>>> G = DiCyclicGroup(Integer(3))
|
[12,1] |
13 |
\(C_{13}\) |
[13,1] |
|
14 |
\(C_{14}\) |
[14,2] |
|
14 |
\(D_{7}\) |
[14,1] |
|
15 |
\(C_{15}\) |
[15,1] |
|
16 |
\(C_{16}\) |
[16,1] |
|
16 |
\(C_8 \times C_2\) |
[16,5] |
|
16 |
\(C_4 \times C_4\) |
[16,2] |
|
16 |
\(C_4\times C_2\times C_2\) |
[16,10] |
|
16 |
\({C_2}^4\) |
[16,14] |
|
16 |
\(D_4 \times C_2\) |
[16,11] |
|
16 |
\(Q \times C_2\) |
[16,12] |
|
16 |
\(D_8\) |
[16,7] |
|
16 |
\(Q_{8}\) （16 阶双环群） |
sage: G = DiCyclicGroup(4)
>>> from sage.all import *
>>> G = DiCyclicGroup(Integer(4))
|
[16,9] |
16 |
\(2^4\) 阶半二面体群 |
sage: G = SemidihedralGroup(4)
>>> from sage.all import *
>>> G = SemidihedralGroup(Integer(4))
|
[16,8] |
16 |
\(2^4\) 阶分裂亚循环群 |
sage: G = SplitMetacyclicGroup(2,4)
>>> from sage.all import *
>>> G = SplitMetacyclicGroup(Integer(2),Integer(4))
|
[16,6] |
16 |
\((C_4 \times C_2) \rtimes_{\phi} C_2\) |
sage: C2 = SymmetricGroup(2); C4 = CyclicPermutationGroup(4)
sage: A = direct_product_permgroups([C2,C4])
sage: alpha = PermutationGroupMorphism(A,A,[A.gens()[0],A.gens()[0]^2*A.gens()[1]])
sage: phi = [[(1,2)],[alpha]]
>>> from sage.all import *
>>> C2 = SymmetricGroup(Integer(2)); C4 = CyclicPermutationGroup(Integer(4))
>>> A = direct_product_permgroups([C2,C4])
>>> alpha = PermutationGroupMorphism(A,A,[A.gens()[Integer(0)],A.gens()[Integer(0)]**Integer(2)*A.gens()[Integer(1)]])
>>> phi = [[(Integer(1),Integer(2))],[alpha]]
|
[16,13] |
16 |
\((C_4 \times C_2) \rtimes_{\phi} C_2\) |
sage: C2 = SymmetricGroup(2); C4 = CyclicPermutationGroup(4)
sage: A = direct_product_permgroups([C2,C4])
sage: alpha = PermutationGroupMorphism(A,A,[A.gens()[0]^3*A.gens()[1],A.gens()[1]])
sage: phi = [[(1,2)],[alpha]]
>>> from sage.all import *
>>> C2 = SymmetricGroup(Integer(2)); C4 = CyclicPermutationGroup(Integer(4))
>>> A = direct_product_permgroups([C2,C4])
>>> alpha = PermutationGroupMorphism(A,A,[A.gens()[Integer(0)]**Integer(3)*A.gens()[Integer(1)],A.gens()[Integer(1)]])
>>> phi = [[(Integer(1),Integer(2))],[alpha]]
|
[16,3] |
16 |
\(C_4 \rtimes_{\phi} C_4\) |
sage: C4 = CyclicPermutationGroup(4)
sage: alpha = PermutationGroupMorphism(C4,C4,[C4.gen().inverse()])
sage: phi = [[(1,2,3,4)],[alpha]]
>>> from sage.all import *
>>> C4 = CyclicPermutationGroup(Integer(4))
>>> alpha = PermutationGroupMorphism(C4,C4,[C4.gen().inverse()])
>>> phi = [[(Integer(1),Integer(2),Integer(3),Integer(4))],[alpha]]
|
[16,4] |
17 |
\(C_{17}\) |
[17,1] |
|
18 |
\(C_{18}\) |
[18,2] |
|
18 |
\(C_6 \times C_3\) |
[18,5] |
|
18 |
\(D_9\) |
[18,1] |
|
18 |
\(S_3 \times C_3\) |
[18,3] |
|
18 |
\(Dih(C_3 \times C_3)\) |
sage: G = GeneralDihedralGroup([3,3])
>>> from sage.all import *
>>> G = GeneralDihedralGroup([Integer(3),Integer(3)])
|
[18,4] |
19 |
\(C_{19}\) |
[19,1] |
|
20 |
\(C_{20}\) |
[20,2] |
|
20 |
\(C_{10} \times C_2\) |
[20,5] |
|
20 |
\(D_{10}\) |
[20,4] |
|
20 |
\(Q_{10}\) （20 阶双环群） |
[20,1] |
|
20 |
\(Hol(C_5)\) |
sage: C5 = CyclicPermutationGroup(5)
sage: G = C5.holomorph()
>>> from sage.all import *
>>> C5 = CyclicPermutationGroup(Integer(5))
>>> G = C5.holomorph()
|
[20,3] |
21 |
\(C_{21}\) |
[21,2] |
|
21 |
\(C_7 \rtimes_{\phi} C_3\) |
sage: C7 = CyclicPermutationGroup(7)
sage: alpha = PermutationGroupMorphism(C7,C7,[C7.gen()**4])
sage: phi = [[(1,2,3)],[alpha]]
>>> from sage.all import *
>>> C7 = CyclicPermutationGroup(Integer(7))
>>> alpha = PermutationGroupMorphism(C7,C7,[C7.gen()**Integer(4)])
>>> phi = [[(Integer(1),Integer(2),Integer(3))],[alpha]]
|
[21,1] |
22 |
\(C_{22}\) |
[22,2] |
|
22 |
\(D_{11}\) |
[22,1] |
|
23 |
\(C_{23}\) |
[23,1] |
|
24 |
\(C_{24}\) |
[24,2] |
|
24 |
\(D_{12}\) |
[24,6] |
|
24 |
\(Q_{12}\) （24 阶双环群） |
[24,4] |
|
24 |
\(C_{12} \times C_2\) |
[24,9] |
|
24 |
\(C_6 \times C_2 \times C_2\) |
[24,15] |
|
24 |
\(S_4\) （四字母对称群） |
sage: G = SymmetricGroup(4)
>>> from sage.all import *
>>> G = SymmetricGroup(Integer(4))
|
[24,12] |
24 |
\(S_3 \times C_4\) |
[24,5] |
|
24 |
\(S_3 \times C_2 \times C_2\) |
[24,14] |
|
24 |
\(D_4 \times C_3\) |
[24,10] |
|
24 |
\(Q \times C_3\) |
[24,11] |
|
24 |
\(A_4 \times C_2\) |
[24,13] |
|
24 |
\(Q_6 \times C_2\) |
[24,7] |
|
24 |
\(Q \rtimes_{\phi} C_3\) |
sage: Q = QuaternionGroup()
sage: alpha = PermutationGroupMorphism(Q,Q,[Q.gens()[0]*Q.gens()[1],Q.gens()[0].inverse()])
sage: phi = [[(1,2,3)],[alpha]]
>>> from sage.all import *
>>> Q = QuaternionGroup()
>>> alpha = PermutationGroupMorphism(Q,Q,[Q.gens()[Integer(0)]*Q.gens()[Integer(1)],Q.gens()[Integer(0)].inverse()])
>>> phi = [[(Integer(1),Integer(2),Integer(3))],[alpha]]
|
[24,3] |
24 |
\(C_3 \rtimes_{\phi} C_8\) |
sage: C3 = CyclicPermutationGroup(3)
sage: alpha = PermutationGroupMorphism(C3,C3,[C3.gen().inverse()])
sage: phi = [[(1,2,3,4,5,6,7,8)],[alpha]]
>>> from sage.all import *
>>> C3 = CyclicPermutationGroup(Integer(3))
>>> alpha = PermutationGroupMorphism(C3,C3,[C3.gen().inverse()])
>>> phi = [[(Integer(1),Integer(2),Integer(3),Integer(4),Integer(5),Integer(6),Integer(7),Integer(8))],[alpha]]
|
[24,1] |
24 |
\(C_3 \rtimes_{\phi} D_4\) |
sage: C3 = CyclicPermutationGroup(3)
sage: alpha1 = PermutationGroupMorphism(C3,C3,[C3.gen().inverse()])
sage: alpha2 = PermutationGroupMorphism(C3,C3,[C3.gen()])
sage: phi = [[(1,2,3,4),(1,3)],[alpha1,alpha2]]
>>> from sage.all import *
>>> C3 = CyclicPermutationGroup(Integer(3))
>>> alpha1 = PermutationGroupMorphism(C3,C3,[C3.gen().inverse()])
>>> alpha2 = PermutationGroupMorphism(C3,C3,[C3.gen()])
>>> phi = [[(Integer(1),Integer(2),Integer(3),Integer(4)),(Integer(1),Integer(3))],[alpha1,alpha2]]
|
[24,8] |
25 |
\(C_{25}\) |
[25,1] |
|
25 |
\(C_5 \times C_5\) |
[25,2] |
|
26 |
\(C_{26}\) |
[26,2] |
|
26 |
\(D_{13}\) |
[26,1] |
|
27 |
\(C_{27}\) |
[27,1] |
|
27 |
\(C_9 \times C_3\) |
[27,2] |
|
27 |
\(C_3 \times C_3 \times C_3\) |
[27,5] |
|
27 |
\(3^3\) 阶分裂亚循环群 |
sage: G = SplitMetacyclicGroup(3,3)
>>> from sage.all import *
>>> G = SplitMetacyclicGroup(Integer(3),Integer(3))
|
[27,4] |
27 |
\((C_3 \times C_3) \rtimes_{\phi} C_3\) |
sage: C3 = CyclicPermutationGroup(3)
sage: A = direct_product_permgroups([C3,C3])
sage: alpha = PermutationGroupMorphism(A,A,[A.gens()[0]*A.gens()[1].inverse(),A.gens()[1]])
sage: phi = [[(1,2,3)],[alpha]]
>>> from sage.all import *
>>> C3 = CyclicPermutationGroup(Integer(3))
>>> A = direct_product_permgroups([C3,C3])
>>> alpha = PermutationGroupMorphism(A,A,[A.gens()[Integer(0)]*A.gens()[Integer(1)].inverse(),A.gens()[Integer(1)]])
>>> phi = [[(Integer(1),Integer(2),Integer(3))],[alpha]]
|
[27,3] |
28 |
\(C_{28}\) |
[28,2] |
|
28 |
\(C_{14} \times C_2\) |
[28,4] |
|
28 |
\(D_{14}\) |
[28,3] |
|
28 |
\(Q_{14}\) （28 阶双环群） |
[28,1] |
|
29 |
\(C_{29}\) |
[29,1] |
|
30 |
\(C_{30}\) |
[30,4] |
|
30 |
\(D_{15}\) |
[30,3] |
|
30 |
\(D_5 \times C_3\) |
[30,2] |
|
30 |
\(D_3 \times C_5\) |
[30,1] |
|
31 |
\(C_{31}\) |
[31,1] |
该表由 Kevin Halasz 提供。
小于等于 15 阶的有限呈示群的构建说明¶
Sage 能够轻松构建阶数小于等于 15 的有限呈示群。 我们将首先探讨创建有限生成阿贝尔群，以及有限呈示群的直积和半直积。
所有有限生成阿贝尔群都可以使用 groups.presentation.FGAbelian(ls) 命令创建，
其中 ls 是一个非负整数列表，该列表被简化为定义要返回群的不变量。
例如，要构建 \(C_4 \times C_2 \times C_2 \times C_2\)，我们可以简单地使用:
sage: A = groups.presentation.FGAbelian([4,2,2,2])
>>> from sage.all import *
>>> A = groups.presentation.FGAbelian([Integer(4),Integer(2),Integer(2),Integer(2)])
无论输入的整数列表如何，对于给定群的输出都是相同的。 以下示例为阶数为 30 的循环群产生相同的表示。
sage: A = groups.presentation.FGAbelian([2,3,5])
sage: B = groups.presentation.FGAbelian([30])
>>> from sage.all import *
>>> A = groups.presentation.FGAbelian([Integer(2),Integer(3),Integer(5)])
>>> B = groups.presentation.FGAbelian([Integer(30)])
如果 G 和 H 是有限呈示群，我们可以使用以下代码来创建 G 和 H 的直积，\(G \times H\)。
sage: D = G.direct_product(H)
>>> from sage.all import *
>>> D = G.direct_product(H)
假设存在从群 \(G\) 到群 \(H\) 的自同构群的同态 \(\phi\)。 通过 \(\phi\) 将 \(G\) 与 \(H\) 的半直积定义为 \(G\) 和 \(H\) 的笛卡尔积， 运算为 \((g_1, h_1)(g_2, h_2) = (g_1 g_2, \phi_{h_1}(g_2) h_2)\) 其中 \(\phi_h = \phi(h)\)。 要在 Sage 中为两个有限呈示群构造此乘积，我们必须使用一对列表手动定义 \(\phi\)。 第一个列表由群 \(G\) 的生成器组成，而第二个列表由第一个列表中相应生成器的像组成。 这些自同构同样定义为一对列表，一个列表为生成器，另一个列表为像。 作为示例，我们将阶数为 16 的二面体群构造为循环群的半直积。
sage: C2 = groups.presentation.Cyclic(2)
sage: C8 = groups.presentation.Cyclic(8)
sage: hom = (C2.gens(), [ ([C8([1])], [C8([-1])]) ])
sage: D = C2.semidirect_product(C8, hom)
>>> from sage.all import *
>>> C2 = groups.presentation.Cyclic(Integer(2))
>>> C8 = groups.presentation.Cyclic(Integer(8))
>>> hom = (C2.gens(), [ ([C8([Integer(1)])], [C8([-Integer(1)])]) ])
>>> D = C2.semidirect_product(C8, hom)
下表显示了阶数小于等于 15 的群，以及如何在 Sage 中构造它们。重复命令已被省略，但通过以下示例进行了描述。
阶数为 \(n\) 的循环群可以通过单个命令创建：
sage: C = groups.presentation.Cyclic(n)
>>> from sage.all import *
>>> C = groups.presentation.Cyclic(n)
对于阶数为 \(2n\) 的二面体群也类似：
sage: D = groups.presentation.Dihedral(n)
>>> from sage.all import *
>>> D = groups.presentation.Dihedral(n)
该表是根据前面 Kevin Halasz 创建的表格构造的。
阶 |
群描述 |
命令 |
GAP ID |
|---|---|---|---|
1 |
平凡群 |
sage: G = groups.presentation.Symmetric(1)
>>> from sage.all import *
>>> G = groups.presentation.Symmetric(Integer(1))
|
[1,1] |
2 |
\(C_2\) |
sage: G = groups.presentation.Symmetric(2)
>>> from sage.all import *
>>> G = groups.presentation.Symmetric(Integer(2))
|
[2,1] |
3 |
\(C_3\) |
sage: G = groups.presentation.Cyclic(3)
>>> from sage.all import *
>>> G = groups.presentation.Cyclic(Integer(3))
|
[3,1] |
4 |
\(C_4\) |
[4,1] |
|
4 |
\(C_2 \times C_2\) |
sage: G = groups.presentation.Klein()
>>> from sage.all import *
>>> G = groups.presentation.Klein()
|
[4,2] |
5 |
\(C_5\) |
[5,1] |
|
6 |
\(C_6\) |
[6,2] |
|
6 |
\(S_3\) （三字母对称群） |
sage: G = groups.presentation.Symmetric(3)
>>> from sage.all import *
>>> G = groups.presentation.Symmetric(Integer(3))
|
[6,1] |
7 |
\(C_7\) |
[7,1] |
|
8 |
\(C_8\) |
[8,1] |
|
8 |
\(C_4 \times C_2\) |
sage: G = groups.presentation.FGAbelian([4,2])
>>> from sage.all import *
>>> G = groups.presentation.FGAbelian([Integer(4),Integer(2)])
|
[8,2] |
8 |
\(C_2\times C_2\times C_2\) |
sage: G = groups.presentation.FGAbelian([2,2,2])
>>> from sage.all import *
>>> G = groups.presentation.FGAbelian([Integer(2),Integer(2),Integer(2)])
|
[8,5] |
8 |
\(D_4\) |
sage: G = groups.presentation.Dihedral(4)
>>> from sage.all import *
>>> G = groups.presentation.Dihedral(Integer(4))
|
[8,3] |
8 |
四元群 (Q) |
sage: G = groups.presentation.Quaternion()
>>> from sage.all import *
>>> G = groups.presentation.Quaternion()
|
[8,4] |
9 |
\(C_9\) |
[9,1] |
|
9 |
\(C_3 \times C_3\) |
[9,2] |
|
10 |
\(C_{10}\) |
[10,2] |
|
10 |
\(D_5\) |
[10,1] |
|
11 |
\(C_{11}\) |
[11,1] |
|
12 |
\(C_{12}\) |
[12,2] |
|
12 |
\(C_6 \times C_2\) |
[12,5] |
|
12 |
\(D_6\) |
[12,4] |
|
12 |
\(A_4\) （四字母交错群） |
sage: G = groups.presentation.Alternating(4)
>>> from sage.all import *
>>> G = groups.presentation.Alternating(Integer(4))
|
[12,3] |
12 |
\(Q_6\) （12 阶双环群） |
sage: G = groups.presentation.DiCyclic(3)
>>> from sage.all import *
>>> G = groups.presentation.DiCyclic(Integer(3))
|
[12,1] |
13 |
\(C_{13}\) |
[13,1] |
|
14 |
\(C_{14}\) |
[14,2] |
|
14 |
\(D_{7}\) |
[14,1] |
|
15 |
\(C_{15}\) |
[15,1] |