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 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] ) sage: GL = groups.matrix.GL(3, ZZ) sage: GL.center() Traceback (most recent call last): ... NotImplementedError: group must be finite
- centralizer(g)#
Return the centralizer of
g
inself
.EXAMPLES:
sage: G = groups.matrix.GL(2, 3) sage: g = G([[1,1], [1,0]]) sage: C = G.centralizer(g); C Subgroup with 3 generators ( [1 1] [2 0] [2 1] [1 0], [0 2], [1 1] ) of General Linear Group of degree 2 over Finite Field of size 3 sage: C.order() 8 sage: S = G.subgroup([G([[2,0],[0,2]]), G([[0,1],[2,0]])]); S Subgroup with 2 generators ( [2 0] [0 1] [0 2], [2 0] ) of General Linear Group of degree 2 over Finite Field of size 3 sage: G.centralizer(S) Subgroup with 3 generators ( [2 0] [0 1] [2 2] [0 2], [2 0], [1 2] ) of General Linear Group of degree 2 over Finite Field of size 3 sage: G = GL(3,2) sage: all(G.order() == G.centralizer(x).order() * G.conjugacy_class(x).cardinality() ....: for x in G) True sage: H = groups.matrix.Heisenberg(2) sage: H.centralizer(H.an_element()) Traceback (most recent call last): ... NotImplementedError: group must be finite
- 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
- 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() # long time [ 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 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
- 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
- conjugacy_classes_subgroups()#
Return a complete list of representatives of conjugacy classes of subgroups in
self
.The ordering is that given by GAP.
EXAMPLES:
sage: G = groups.matrix.GL(2,2) sage: G.conjugacy_classes_subgroups() [Subgroup with 0 generators () of General Linear Group of degree 2 over Finite Field of size 2, Subgroup with 1 generators ( [1 1] [0 1] ) of General Linear Group of degree 2 over Finite Field of size 2, Subgroup with 1 generators ( [0 1] [1 1] ) of General Linear Group of degree 2 over Finite Field of size 2, Subgroup with 2 generators ( [0 1] [1 1] [1 1], [0 1] ) of General Linear Group of degree 2 over Finite Field of size 2] sage: H = groups.matrix.Heisenberg(2) sage: H.conjugacy_classes_subgroups() Traceback (most recent call last): ... NotImplementedError: group must be finite
- exponent()#
Computes the exponent of the group.
The exponent \(e\) of a group \(G\) is the LCM of the orders of its elements, that is, \(e\) is the smallest integer such that \(g^e = 1\) for all \(g \in G\).
EXAMPLES:
sage: G = groups.matrix.GL(2, 3) sage: G.exponent() 24 sage: H = groups.matrix.Heisenberg(2) sage: H.exponent() Traceback (most recent call last): ... NotImplementedError: group must be finite
- group_id()#
Return the ID code of
self
, which is a list of two integers.It is a unique identified assigned by GAP for groups in the
SmallGroup
library.EXAMPLES:
sage: PGL(2,3).group_id() [24, 12] sage: SymmetricGroup(4).group_id() [24, 12] sage: G = groups.matrix.GL(2, 2) sage: G.group_id() [6, 1] sage: G = groups.matrix.GL(2, 3) sage: G.id() [48, 29] sage: G = groups.matrix.GL(2, ZZ) sage: G.group_id() Traceback (most recent call last): ... GAPError: Error, the group identification for groups of size infinity is not available
- id()#
Return the ID code of
self
, which is a list of two integers.It is a unique identified assigned by GAP for groups in the
SmallGroup
library.EXAMPLES:
sage: PGL(2,3).group_id() [24, 12] sage: SymmetricGroup(4).group_id() [24, 12] sage: G = groups.matrix.GL(2, 2) sage: G.group_id() [6, 1] sage: G = groups.matrix.GL(2, 3) sage: G.id() [48, 29] sage: G = groups.matrix.GL(2, ZZ) sage: G.group_id() Traceback (most recent call last): ... GAPError: Error, the group identification for groups of size infinity is not available
- 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()#
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
- 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
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)
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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 github issue #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 github issue #10515):
sage: gens = [matrix(QQ,2,[1,0,0,1])] sage: MatrixGroup(gens).list() ( [1 0] [0 1] )
Another example over a ring (see github issue #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
- subgroups()#
Return a list of all the subgroups of
self
.OUTPUT:
Each possible subgroup of
self
is contained once in the returned list. The list is in order, according to the size of the subgroups, from the trivial subgroup with one element on through up to the whole group. Conjugacy classes of subgroups are contiguous in the list.Warning
For even relatively small groups this method can take a very long time to execute, or create vast amounts of output. Likely both. Its purpose is instructional, as it can be useful for studying small groups.
For faster results, which still exhibit the structure of the possible subgroups, use
conjugacy_classes_subgroups()
.EXAMPLES:
sage: G = groups.matrix.GL(2, 2) sage: G.subgroups() [Subgroup with 0 generators () of General Linear Group of degree 2 over Finite Field of size 2, Subgroup with 1 generators ( [0 1] [1 0] ) of General Linear Group of degree 2 over Finite Field of size 2, Subgroup with 1 generators ( [1 0] [1 1] ) of General Linear Group of degree 2 over Finite Field of size 2, Subgroup with 1 generators ( [1 1] [0 1] ) of General Linear Group of degree 2 over Finite Field of size 2, Subgroup with 1 generators ( [0 1] [1 1] ) of General Linear Group of degree 2 over Finite Field of size 2, Subgroup with 2 generators ( [0 1] [1 1] [1 1], [0 1] ) of General Linear Group of degree 2 over Finite Field of size 2] sage: H = groups.matrix.Heisenberg(2) sage: H.subgroups() Traceback (most recent call last): ... NotImplementedError: group must be finite
- 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