Endliche und abelsche Gruppen¶
Sage unterstützt das Rechnen mit Permutationsgruppen, endlichen klassischen Gruppen (wie z.B. \(SU(n,q)\)), endlichen Matrixgruppen (mit Ihren eigenen Erzeugern), und abelschen Gruppen (sogar unendlichen). Vieles davon ist mit Hilfe der GAP-Schnittstelle implementiert.
Zum Beispiel können Sie, um eine Permutationsgruppe zu erzeugen, die Liste der Erzeuger wie folgt angeben.
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() # random-ish output
[Permutation Group with generators [(1,2,3)(4,5), (3,4)],
Permutation Group with generators [(1,5)(3,4), (1,5)(2,4), (1,3,5)]]
sage: G.center()
Subgroup generated by [()] of (Permutation Group with generators [(3,4), (1,2,3)(4,5)])
sage: G.random_element() # random output
(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() # random-ish output
[Permutation Group with generators [(1,2,3)(4,5), (3,4)],
Permutation Group with generators [(1,5)(3,4), (1,5)(2,4), (1,3,5)]]
>>> G.center()
Subgroup generated by [()] of (Permutation Group with generators [(3,4), (1,2,3)(4,5)])
>>> G.random_element() # random output
(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() # random-ish output G.center() G.random_element() # random output print(latex(G))
Sie können in Sage auch die Tabelle der Charaktere (im LaTeX-Format) erhalten:
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 beinhaltet auch klassische und Matrixgruppen über endlichen Körpern:
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 output
[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 output
[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 output G.order()
Sie können auch mit (endlichen oder unendlichen) abelschen Gruppen rechnen.
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()