Mix-in Class for libGAP-based Groups

This class adds access to GAP functionality to groups such that parent and element have a gap() method that returns a libGAP object for the parent/element.

If your group implementation uses libgap, then you should add GroupMixinLibGAP as the first class that you are deriving from. This ensures that it properly overrides any default methods that just raise NotImplemented.

class sage.groups.libgap_mixin.GroupMixinLibGAP

Bases: object

cardinality()

Implements EnumeratedSets.ParentMethods.cardinality().

EXAMPLES:

sage: G = Sp(4,GF(3))
sage: G.cardinality()
51840

sage: G = SL(4,GF(3))
sage: G.cardinality()
12130560

sage: F = GF(5); MS = MatrixSpace(F,2,2)
sage: gens = [MS([[1,2],[-1,1]]),MS([[1,1],[0,1]])]
sage: G = MatrixGroup(gens)
sage: G.cardinality()
480

sage: G = MatrixGroup([matrix(ZZ,2,[1,1,0,1])])
sage: G.cardinality()
+Infinity

sage: G = Sp(4,GF(3))
sage: G.cardinality()
51840

sage: G = SL(4,GF(3))
sage: G.cardinality()
12130560

sage: F = GF(5); MS = MatrixSpace(F,2,2)
sage: gens = [MS([[1,2],[-1,1]]),MS([[1,1],[0,1]])]
sage: G = MatrixGroup(gens)
sage: G.cardinality()
480

sage: G = MatrixGroup([matrix(ZZ,2,[1,1,0,1])])
sage: G.cardinality()
+Infinity
center()

Return the center of this linear group as a subgroup.

OUTPUT:

The center as a subgroup.

EXAMPLES:

sage: G = SU(3,GF(2))
sage: G.center()
Matrix group over Finite Field in a of size 2^2 with 1 generators (
[a 0 0]
[0 a 0]
[0 0 a]
)
sage: GL(2,GF(3)).center()
Matrix group over Finite Field of size 3 with 1 generators (
[2 0]
[0 2]
)
sage: GL(3,GF(3)).center()
Matrix group over Finite Field of size 3 with 1 generators (
[2 0 0]
[0 2 0]
[0 0 2]
)
sage: GU(3,GF(2)).center()
Matrix group over Finite Field in a of size 2^2 with 1 generators (
[a + 1     0     0]
[    0 a + 1     0]
[    0     0 a + 1]
)

sage: A = Matrix(FiniteField(5), [[2,0,0], [0,3,0], [0,0,1]])
sage: B = Matrix(FiniteField(5), [[1,0,0], [0,1,0], [0,1,1]])
sage: MatrixGroup([A,B]).center()
Matrix group over Finite Field of size 5 with 1 generators (
[1 0 0]
[0 1 0]
[0 0 1]
)
class_function(values)

Return the class function with given values.

INPUT:

  • values – list/tuple/iterable of numbers. The values of the class function on the conjugacy classes, in that order.

EXAMPLES:

sage: G = GL(2,GF(3))
sage: chi = G.class_function(range(8))
sage: list(chi)
[0, 1, 2, 3, 4, 5, 6, 7]
conjugacy_class(g)

Return the conjugacy class of g.

OUTPUT:

The conjugacy class of g in the group self. If self is the group denoted by \(G\), this method computes the set \(\{x^{-1}gx\ \vert\ x\in G\}\).

EXAMPLES:

sage: G = SL(2, QQ)
sage: g = G([[1,1],[0,1]])
sage: G.conjugacy_class(g)
Conjugacy class of [1 1]
[0 1] in Special Linear Group of degree 2 over Rational Field
conjugacy_class_representatives()

Return a set of representatives for each of the conjugacy classes of the group.

EXAMPLES:

sage: G = SU(3,GF(2))
sage: len(G.conjugacy_class_representatives())
16

sage: G = GL(2,GF(3))
sage: G.conjugacy_class_representatives()
(
[1 0]  [0 2]  [2 0]  [0 2]  [0 2]  [0 1]  [0 1]  [2 0]
[0 1], [1 1], [0 2], [1 2], [1 0], [1 2], [1 1], [0 1]
)

sage: len(GU(2,GF(5)).conjugacy_class_representatives())
36
conjugacy_classes()

Return a list with all the conjugacy classes of self.

EXAMPLES:

sage: G = SL(2, GF(2))
sage: G.conjugacy_classes()
(Conjugacy class of [1 0]
 [0 1] in Special Linear Group of degree 2 over Finite Field of size 2,
 Conjugacy class of [0 1]
 [1 0] in Special Linear Group of degree 2 over Finite Field of size 2,
 Conjugacy class of [0 1]
 [1 1] in Special Linear Group of degree 2 over Finite Field of size 2)
intersection(other)

Return the intersection of two groups (if it makes sense) as a subgroup of the first group.

EXAMPLES:

sage: A = Matrix([(0, 1/2, 0), (2, 0, 0), (0, 0, 1)])
sage: B = Matrix([(0, 1/2, 0), (-2, -1, 2), (0, 0, 1)])
sage: G = MatrixGroup([A,B])
sage: len(G)  # isomorphic to S_3
6
sage: G.intersection(GL(3,ZZ))
Matrix group over Rational Field with 1 generators (
[ 1  0  0]
[-2 -1  2]
[ 0  0  1]
)
sage: GL(3,ZZ).intersection(G)
Matrix group over Integer Ring with 1 generators (
[ 1  0  0]
[-2 -1  2]
[ 0  0  1]
)
sage: G.intersection(SL(3,ZZ))
Matrix group over Rational Field with 0 generators ()
irreducible_characters()

Returns the irreducible characters of the group.

OUTPUT:

A tuple containing all irreducible characters.

EXAMPLES:

sage: G = GL(2,2)
sage: G.irreducible_characters()
(Character of General Linear Group of degree 2 over Finite Field of size 2,
 Character of General Linear Group of degree 2 over Finite Field of size 2,
 Character of General Linear Group of degree 2 over Finite Field of size 2)
is_abelian()

Test whether the group is Abelian.

OUTPUT:

Boolean. True if this group is an Abelian group.

EXAMPLES:

sage: SL(1, 17).is_abelian()
True
sage: SL(2, 17).is_abelian()
False
is_finite()

Test whether the matrix group is finite.

OUTPUT:

Boolean.

EXAMPLES:

sage: G = GL(2,GF(3))
sage: G.is_finite()
True
sage: SL(2,ZZ).is_finite()
False
is_isomorphic(H)

Test whether self and H are isomorphic groups.

INPUT:

  • H – a group.

OUTPUT:

Boolean.

EXAMPLES:

sage: m1 = matrix(GF(3), [[1,1],[0,1]])
sage: m2 = matrix(GF(3), [[1,2],[0,1]])
sage: F = MatrixGroup(m1)
sage: G = MatrixGroup(m1, m2)
sage: H = MatrixGroup(m2)
sage: F.is_isomorphic(G)
True
sage: G.is_isomorphic(H)
True
sage: F.is_isomorphic(H)
True
sage: F==G, G==H, F==H
(False, False, False)
list()

List all elements of this group.

OUTPUT:

A tuple containing all group elements in a random but fixed order.

EXAMPLES:

sage: F = GF(3)
sage: gens = [matrix(F,2, [1,0, -1,1]), matrix(F, 2, [1,1,0,1])]
sage: G = MatrixGroup(gens)
sage: G.cardinality()
24
sage: v = G.list()
sage: len(v)
24
sage: v[:5]
(
[1 0]  [2 0]  [0 1]  [0 2]  [1 2]
[0 1], [0 2], [2 0], [1 0], [2 2]
)
sage: all(g in G for g in G.list())
True

An example over a ring (see trac ticket #5241):

sage: M1 = matrix(ZZ,2,[[-1,0],[0,1]])
sage: M2 = matrix(ZZ,2,[[1,0],[0,-1]])
sage: M3 = matrix(ZZ,2,[[-1,0],[0,-1]])
sage: MG = MatrixGroup([M1, M2, M3])
sage: MG.list()
(
[1 0]  [ 1  0]  [-1  0]  [-1  0]
[0 1], [ 0 -1], [ 0  1], [ 0 -1]
)
sage: MG.list()[1]
[ 1  0]
[ 0 -1]
sage: MG.list()[1].parent()
Matrix group over Integer Ring with 3 generators (
[-1  0]  [ 1  0]  [-1  0]
[ 0  1], [ 0 -1], [ 0 -1]
)

An example over a field (see trac ticket #10515):

sage: gens = [matrix(QQ,2,[1,0,0,1])]
sage: MatrixGroup(gens).list()
(
[1 0]
[0 1]
)

Another example over a ring (see trac ticket #9437):

sage: len(SL(2, Zmod(4)).list())
48

An error is raised if the group is not finite:

sage: GL(2,ZZ).list()
Traceback (most recent call last):
...
NotImplementedError: group must be finite
order()

Implements EnumeratedSets.ParentMethods.cardinality().

EXAMPLES:

sage: G = Sp(4,GF(3))
sage: G.cardinality()
51840

sage: G = SL(4,GF(3))
sage: G.cardinality()
12130560

sage: F = GF(5); MS = MatrixSpace(F,2,2)
sage: gens = [MS([[1,2],[-1,1]]),MS([[1,1],[0,1]])]
sage: G = MatrixGroup(gens)
sage: G.cardinality()
480

sage: G = MatrixGroup([matrix(ZZ,2,[1,1,0,1])])
sage: G.cardinality()
+Infinity

sage: G = Sp(4,GF(3))
sage: G.cardinality()
51840

sage: G = SL(4,GF(3))
sage: G.cardinality()
12130560

sage: F = GF(5); MS = MatrixSpace(F,2,2)
sage: gens = [MS([[1,2],[-1,1]]),MS([[1,1],[0,1]])]
sage: G = MatrixGroup(gens)
sage: G.cardinality()
480

sage: G = MatrixGroup([matrix(ZZ,2,[1,1,0,1])])
sage: G.cardinality()
+Infinity
random_element()

Return a random element of this group.

OUTPUT:

A group element.

EXAMPLES:

sage: G = Sp(4,GF(3))
sage: G.random_element()  # random
[2 1 1 1]
[1 0 2 1]
[0 1 1 0]
[1 0 0 1]
sage: G.random_element() in G
True

sage: F = GF(5); MS = MatrixSpace(F,2,2)
sage: gens = [MS([[1,2],[-1,1]]),MS([[1,1],[0,1]])]
sage: G = MatrixGroup(gens)
sage: G.random_element()  # random
[1 3]
[0 3]
sage: G.random_element() in G
True