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
.
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class
sage.groups.libgap_mixin.
GroupMixinLibGAP
¶ Bases:
object
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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
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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] )
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character
(values)¶ Return a group character from
values
, wherevalues
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
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character_table
()¶ Return 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]
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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]
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conjugacy_class
(g)¶ Return the conjugacy class of
g
.OUTPUT:
The conjugacy class of
g
in the groupself
. Ifself
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
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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
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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
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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] )
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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
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is_abelian
()¶ Return whether the group is Abelian.
OUTPUT:
Boolean.
True
if this group is an Abelian group andFalse
otherwise.EXAMPLES:
sage: from sage.groups.libgap_group import GroupLibGAP sage: GroupLibGAP(libgap.CyclicGroup(12)).is_abelian() True sage: GroupLibGAP(libgap.SymmetricGroup(12)).is_abelian() False sage: SL(1, 17).is_abelian() True sage: SL(2, 17).is_abelian() False
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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
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is_isomorphic
(H)¶ Test whether
self
andH
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)
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is_nilpotent
()¶ Return whether this group is nilpotent.
EXAMPLES:
sage: from sage.groups.libgap_group import GroupLibGAP sage: GroupLibGAP(libgap.AlternatingGroup(3)).is_nilpotent() True sage: GroupLibGAP(libgap.SymmetricGroup(3)).is_nilpotent() False
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is_p_group
()¶ Return whether this group is a p-group.
EXAMPLES:
sage: from sage.groups.libgap_group import GroupLibGAP sage: GroupLibGAP(libgap.CyclicGroup(9)).is_p_group() True sage: GroupLibGAP(libgap.CyclicGroup(10)).is_p_group() False
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is_perfect
()¶ Return whether this group is perfect.
EXAMPLES:
sage: from sage.groups.libgap_group import GroupLibGAP sage: GroupLibGAP(libgap.SymmetricGroup(5)).is_perfect() False sage: GroupLibGAP(libgap.AlternatingGroup(5)).is_perfect() True sage: SL(3,3).is_perfect() True
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is_polycyclic
()¶ Return whether this group is polycyclic.
EXAMPLES:
sage: from sage.groups.libgap_group import GroupLibGAP sage: GroupLibGAP(libgap.AlternatingGroup(4)).is_polycyclic() True sage: GroupLibGAP(libgap.AlternatingGroup(5)).is_solvable() False
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is_simple
()¶ Return whether this group is simple.
EXAMPLES:
sage: from sage.groups.libgap_group import GroupLibGAP sage: GroupLibGAP(libgap.SL(2,3)).is_simple() False sage: GroupLibGAP(libgap.SL(3,3)).is_simple() True sage: SL(3,3).is_simple() True
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is_solvable
()¶ Return whether this group is solvable.
EXAMPLES:
sage: from sage.groups.libgap_group import GroupLibGAP sage: GroupLibGAP(libgap.SymmetricGroup(4)).is_solvable() True sage: GroupLibGAP(libgap.SymmetricGroup(5)).is_solvable() False
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is_supersolvable
()¶ Return whether this group is supersolvable.
EXAMPLES:
sage: from sage.groups.libgap_group import GroupLibGAP sage: GroupLibGAP(libgap.SymmetricGroup(3)).is_supersolvable() True sage: GroupLibGAP(libgap.SymmetricGroup(4)).is_supersolvable() False
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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
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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
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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
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trivial_character
()¶ Return 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
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