有限群，アーベル群

Sageでは，置換群，有限古典群(例えば \(SU(n,q)\))，有限行列群(生成元を指定して生成)，そしてアーベル群(無限次も可)などの演算が可能である． これらの機能の大半は，GAPとのインターフェイスを経由して実現されている．

まず，例として置換群を生成してみよう． それには，以下のようにして生成元のリストを指定してやればよい．

sage: G = PermutationGroup(['(1,2,3)(4,5)', '(3,4)'])
sage: G
Permutation Group with generators [(3,4), (1,2,3)(4,5)]
sage: G.order()
120
sage: G.is_abelian()
False
sage: G.derived_series()           # 結果は変化しがち
[Subgroup generated by [(3,4), (1,2,3)(4,5)] of (Permutation Group with generators [(3,4), (1,2,3)(4,5)]),
 Subgroup generated by [...] of (Permutation Group with generators [(3,4), (1,2,3)(4,5)])]
sage: G.center()
Subgroup generated by [()] of (Permutation Group with generators [(3,4), (1,2,3)(4,5)])
sage: G.random_element()           # random 出力は変化する
(1,5,3)(2,4)
sage: print(latex(G))
\langle (3,4), (1,2,3)(4,5) \rangle

>>> from sage.all import *
>>> G = PermutationGroup(['(1,2,3)(4,5)', '(3,4)'])
>>> G
Permutation Group with generators [(3,4), (1,2,3)(4,5)]
>>> G.order()
120
>>> G.is_abelian()
False
>>> G.derived_series()           # 結果は変化しがち
[Subgroup generated by [(3,4), (1,2,3)(4,5)] of (Permutation Group with generators [(3,4), (1,2,3)(4,5)]),
 Subgroup generated by [...] of (Permutation Group with generators [(3,4), (1,2,3)(4,5)])]
>>> G.center()
Subgroup generated by [()] of (Permutation Group with generators [(3,4), (1,2,3)(4,5)])
>>> G.random_element()           # random 出力は変化する
(1,5,3)(2,4)
>>> print(latex(G))
\langle (3,4), (1,2,3)(4,5) \rangle

G = PermutationGroup(['(1,2,3)(4,5)', '(3,4)'])
G
G.order()
G.is_abelian()
G.derived_series()           # 結果は変化しがち
G.center()
G.random_element()           # random 出力は変化する
print(latex(G))

Sageを使えば(LaTeX形式で)指標表を作ることもできる:

sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3)]])
sage: latex(G.character_table())
\left(\begin{array}{rrrr}
1 & 1 & 1 & 1 \\
1 & -\zeta_{3} - 1 & \zeta_{3} & 1 \\
1 & \zeta_{3} & -\zeta_{3} - 1 & 1 \\
3 & 0 & 0 & -1
\end{array}\right)

>>> from sage.all import *
>>> G = PermutationGroup([[(Integer(1),Integer(2)),(Integer(3),Integer(4))], [(Integer(1),Integer(2),Integer(3))]])
>>> latex(G.character_table())
\left(\begin{array}{rrrr}
1 & 1 & 1 & 1 \\
1 & -\zeta_{3} - 1 & \zeta_{3} & 1 \\
1 & \zeta_{3} & -\zeta_{3} - 1 & 1 \\
3 & 0 & 0 & -1
\end{array}\right)

G = PermutationGroup([[(1,2),(3,4)], [(1,2,3)]])
latex(G.character_table())

Sageは有限体上の古典群と行列群も扱うことができる:

sage: MS = MatrixSpace(GF(7), 2)
sage: gens = [MS([[1,0],[-1,1]]),MS([[1,1],[0,1]])]
sage: G = MatrixGroup(gens)
sage: G.conjugacy_classes_representatives()
(
[1 0]  [0 6]  [0 4]  [6 0]  [0 6]  [0 4]  [0 6]  [0 6]  [0 6]  [4 0]
[0 1], [1 5], [5 5], [0 6], [1 2], [5 2], [1 0], [1 4], [1 3], [0 2],

[5 0]
[0 3]
)
sage: G = Sp(4,GF(7))
sage: G
Symplectic Group of degree 4 over Finite Field of size 7
sage: G.random_element()             # random 元をランダムに出力
[5 5 5 1]
[0 2 6 3]
[5 0 1 0]
[4 6 3 4]
sage: G.order()
276595200

>>> from sage.all import *
>>> MS = MatrixSpace(GF(Integer(7)), Integer(2))
>>> gens = [MS([[Integer(1),Integer(0)],[-Integer(1),Integer(1)]]),MS([[Integer(1),Integer(1)],[Integer(0),Integer(1)]])]
>>> G = MatrixGroup(gens)
>>> G.conjugacy_classes_representatives()
(
[1 0]  [0 6]  [0 4]  [6 0]  [0 6]  [0 4]  [0 6]  [0 6]  [0 6]  [4 0]
[0 1], [1 5], [5 5], [0 6], [1 2], [5 2], [1 0], [1 4], [1 3], [0 2],
<BLANKLINE>
[5 0]
[0 3]
)
>>> G = Sp(Integer(4),GF(Integer(7)))
>>> G
Symplectic Group of degree 4 over Finite Field of size 7
>>> G.random_element()             # random 元をランダムに出力
[5 5 5 1]
[0 2 6 3]
[5 0 1 0]
[4 6 3 4]
>>> G.order()
276595200

MS = MatrixSpace(GF(7), 2)
gens = [MS([[1,0],[-1,1]]),MS([[1,1],[0,1]])]
G = MatrixGroup(gens)
G.conjugacy_classes_representatives()
G = Sp(4,GF(7))
G
G.random_element()             # random 元をランダムに出力
G.order()

(無限次および有限次の)アーベル群を使う演算も可能だ:

sage: F = AbelianGroup(5, [5,5,7,8,9], names='abcde')
sage: (a, b, c, d, e) = F.gens()
sage: d * b**2 * c**3
b^2*c^3*d
sage: F = AbelianGroup(3,[2]*3); F
Multiplicative Abelian group isomorphic to C2 x C2 x C2
sage: H = AbelianGroup([2,3], names="xy"); H
Multiplicative Abelian group isomorphic to C2 x C3
sage: AbelianGroup(5)
Multiplicative Abelian group isomorphic to Z x Z x Z x Z x Z
sage: AbelianGroup(5).order()
+Infinity

>>> from sage.all import *
>>> F = AbelianGroup(Integer(5), [Integer(5),Integer(5),Integer(7),Integer(8),Integer(9)], names='abcde')
>>> (a, b, c, d, e) = F.gens()
>>> d * b**Integer(2) * c**Integer(3)
b^2*c^3*d
>>> F = AbelianGroup(Integer(3),[Integer(2)]*Integer(3)); F
Multiplicative Abelian group isomorphic to C2 x C2 x C2
>>> H = AbelianGroup([Integer(2),Integer(3)], names="xy"); H
Multiplicative Abelian group isomorphic to C2 x C3
>>> AbelianGroup(Integer(5))
Multiplicative Abelian group isomorphic to Z x Z x Z x Z x Z
>>> AbelianGroup(Integer(5)).order()
+Infinity

F = AbelianGroup(5, [5,5,7,8,9], names='abcde')
(a, b, c, d, e) = F.gens()
d * b**2 * c**3
F = AbelianGroup(3,[2]*3); F
H = AbelianGroup([2,3], names="xy"); H
AbelianGroup(5)
AbelianGroup(5).order()