Mix-in Class for GAP-based Groups¶

This class adds access to GAP functionality to groups such that parent and element have a gap() method that returns a GAP 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 NotImplementedError.

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()
Subgroup with 1 generators (
[a 0 0]
[0 a 0]
[0 0 a]
) of Special Unitary Group of degree 3 over Finite Field in a of size 2^2
sage: GL(2,GF(3)).center()
Subgroup with 1 generators (
[2 0]
[0 2]
) of General Linear Group of degree 2 over Finite Field of size 3
sage: GL(3,GF(3)).center()
Subgroup with 1 generators (
[2 0 0]
[0 2 0]
[0 0 2]
) of General Linear Group of degree 3 over Finite Field of size 3
sage: GU(3,GF(2)).center()
Subgroup with 1 generators (
[a + 1     0     0]
[    0 a + 1     0]
[    0     0 a + 1]
) of General Unitary Group of degree 3 over Finite Field in a of size 2^2

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()
Subgroup with 1 generators (
[1 0 0]
[0 1 0]
[0 0 1]
) of Matrix group over Finite Field of size 5 with 2 generators (
[2 0 0]  [1 0 0]
[0 3 0]  [0 1 0]
[0 0 1], [0 1 1]
)

character(values)

Returns a group character from values, where values is a list of the values of the character evaluated on the conjugacy classes.

INPUT:

• values – a list of values of the character

OUTPUT: a group character

EXAMPLES:

sage: G = MatrixGroup(AlternatingGroup(4))
sage: G.character([1]*len(G.conjugacy_classes_representatives()))
Character of Matrix group over Integer Ring with 12 generators

sage: G = GL(2,ZZ)
sage: G.character([1,1,1,1])
Traceback (most recent call last):
...
NotImplementedError: only implemented for finite groups

character_table()

Returns the matrix of values of the irreducible characters of this group $$G$$ at its conjugacy classes.

The columns represent the conjugacy classes of $$G$$ and the rows represent the different irreducible characters in the ordering given by GAP.

OUTPUT: a matrix defined over a cyclotomic field

EXAMPLES:

sage: MatrixGroup(SymmetricGroup(2)).character_table()
[ 1 -1]
[ 1  1]
sage: MatrixGroup(SymmetricGroup(3)).character_table()
[ 1  1 -1]
[ 2 -1  0]
[ 1  1  1]
sage: MatrixGroup(SymmetricGroup(5)).character_table()
[ 1 -1 -1  1 -1  1  1]
[ 4  0  1 -1 -2  1  0]
[ 5  1 -1  0 -1 -1  1]
[ 6  0  0  1  0  0 -2]
[ 5 -1  1  0  1 -1  1]
[ 4  0 -1 -1  2  1  0]
[ 1  1  1  1  1  1  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_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)

sage: GL(2,ZZ).conjugacy_classes()
Traceback (most recent call last):
...
NotImplementedError: only implemented for finite groups

conjugacy_classes_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_classes_representatives())
16

sage: G = GL(2,GF(3))
sage: G.conjugacy_classes_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_classes_representatives())
36

sage: GL(2,ZZ).conjugacy_classes_representatives()
Traceback (most recent call last):
...
NotImplementedError: only implemented for finite groups

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))
Subgroup with 1 generators (
[ 1  0  0]
[-2 -1  2]
[ 0  0  1]
) of Matrix group over Rational Field with 2 generators (
[  0 1/2   0]  [  0 1/2   0]
[  2   0   0]  [ -2  -1   2]
[  0   0   1], [  0   0   1]
)
sage: GL(3,ZZ).intersection(G)
Subgroup with 1 generators (
[ 1  0  0]
[-2 -1  2]
[ 0  0  1]
) of General Linear Group of degree 3 over Integer Ring
sage: G.intersection(SL(3,ZZ))
Subgroup with 0 generators () of Matrix group over Rational Field with 2 generators (
[  0 1/2   0]  [  0 1/2   0]
[  2   0   0]  [ -2  -1   2]
[  0   0   1], [  0   0   1]
)

irreducible_characters()

Return 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)

sage: GL(2,ZZ).irreducible_characters()
Traceback (most recent call last):
...
NotImplementedError: only implemented for finite groups

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]
(
[0 1]  [0 1]  [0 1]  [0 2]  [0 2]
[2 0], [2 1], [2 2], [1 0], [1 1]
)
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

trivial_character()

Returns the trivial character of this group.

OUTPUT: a group character

EXAMPLES:

sage: MatrixGroup(SymmetricGroup(3)).trivial_character()
Character of Matrix group over Integer Ring with 6 generators

sage: GL(2,ZZ).trivial_character()
Traceback (most recent call last):
...
NotImplementedError: only implemented for finite groups