EXAMPLES:
sage: G = Sp(4,GF(7)); G
Symplectic Group of degree 4 over Finite Field of size 7
sage: g = prod(G.gens()); g
[3 0 3 0]
[1 0 0 0]
[0 1 0 1]
[0 2 0 0]
sage: m = g.matrix()
sage: m * G.invariant_form() * m.transpose() == G.invariant_form()
True
sage: G.order()
276595200
AUTHORS:
Return the symplectic group.
The special linear group consists of all
matrices that are invertible over the ring
with determinant
one.
Note
This group is also available via groups.matrix.Sp().
INPUT:
EXAMPLES:
sage: Sp(4, 5)
Symplectic Group of degree 4 over Finite Field of size 5
sage: Sp(4, IntegerModRing(15))
Symplectic Group of degree 4 over Ring of integers modulo 15
sage: Sp(3, GF(7))
Traceback (most recent call last):
...
ValueError: the degree must be even
TESTS:
sage: groups.matrix.Sp(2, 3)
Symplectic Group of degree 2 over Finite Field of size 3
sage: G = Sp(4,5)
sage: TestSuite(G).run()
Bases: sage.groups.matrix_gps.symplectic.SymplecticMatrixGroup_generic, sage.groups.matrix_gps.named_group.NamedMatrixGroup_gap
Symplectic group in GAP
EXAMPLES:
sage: Sp(2,4)
Symplectic Group of degree 2 over Finite Field in a of size 2^2
sage: latex(Sp(4,5))
\text{Sp}_{4}(\Bold{F}_{5})
Return the quadratic form preserved by the orthogonal group.
OUTPUT:
A matrix.
EXAMPLES:
sage: Sp(4, GF(3)).invariant_form()
[0 0 0 1]
[0 0 1 0]
[0 2 0 0]
[2 0 0 0]
Bases: sage.groups.matrix_gps.named_group.NamedMatrixGroup_generic
Base class for “named” matrix groups
INPUT:
EXAMPLES:
sage: G = GL(2, QQ)
sage: from sage.groups.matrix_gps.named_group import NamedMatrixGroup_generic
sage: isinstance(G, NamedMatrixGroup_generic)
True
Return the quadratic form preserved by the orthogonal group.
OUTPUT:
A matrix.
EXAMPLES:
sage: Sp(4, QQ).invariant_form()
[0 0 0 1]
[0 0 1 0]
[0 1 0 0]
[1 0 0 0]