LibGAP-based Groups#
This module provides helper class for wrapping GAP groups via
libgap
. See free_group
for an
example how they are used.
The parent class keeps track of the GAP element object, to use it
in your Python parent you have to derive both from the suitable group
parent and ParentLibGAP
sage: from sage.groups.libgap_wrapper import ElementLibGAP, ParentLibGAP
sage: from sage.groups.group import Group
sage: class FooElement(ElementLibGAP):
....: pass
sage: class FooGroup(Group, ParentLibGAP):
....: Element = FooElement
....: def __init__(self):
....: lg = libgap(libgap.CyclicGroup(3)) # dummy
....: ParentLibGAP.__init__(self, lg)
....: Group.__init__(self)
Note how we call the constructor of both superclasses to initialize
Group
and ParentLibGAP
separately. The parent class implements
its output via LibGAP:
sage: FooGroup()
<pc group of size 3 with 1 generators>
sage: type(FooGroup().gap())
<class 'sage.libs.gap.element.GapElement'>
The element class is a subclass of
MultiplicativeGroupElement
. To use
it, you just inherit from ElementLibGAP
sage: element = FooGroup().an_element()
sage: element
f1
The element class implements group operations and printing via LibGAP:
sage: element._repr_()
'f1'
sage: element * element
f1^2
AUTHORS:
Volker Braun
- class sage.groups.libgap_wrapper.ElementLibGAP#
Bases:
MultiplicativeGroupElement
A class for LibGAP-based Sage group elements
INPUT:
parent
– the Sage parentlibgap_element
– the libgap element that is being wrapped
EXAMPLES:
sage: from sage.groups.libgap_wrapper import ElementLibGAP, ParentLibGAP sage: from sage.groups.group import Group sage: class FooElement(ElementLibGAP): ....: pass sage: class FooGroup(Group, ParentLibGAP): ....: Element = FooElement ....: def __init__(self): ....: lg = libgap(libgap.CyclicGroup(3)) # dummy ....: ParentLibGAP.__init__(self, lg) ....: Group.__init__(self) sage: FooGroup() <pc group of size 3 with 1 generators> sage: FooGroup().gens() (f1,)
- gap()#
Return a LibGAP representation of the element.
OUTPUT:
EXAMPLES:
sage: G.<a,b> = FreeGroup('a, b') sage: x = G([1, 2, -1, -2]) sage: x a*b*a^-1*b^-1 sage: xg = x.gap() sage: xg a*b*a^-1*b^-1 sage: type(xg) <class 'sage.libs.gap.element.GapElement'>
- inverse()#
Return the inverse of self.
- is_conjugate(other)#
Return whether the elements
self
andother
are conjugate.EXAMPLES:
sage: from sage.groups.libgap_group import GroupLibGAP sage: G = GroupLibGAP(libgap.GL(2, 3)) sage: a,b = G.gens() sage: a.is_conjugate(b) False sage: a.is_conjugate((a*b^2) * a * ~(a*b^2)) True
- is_one()#
Test whether the group element is the trivial element.
OUTPUT:
Boolean.
EXAMPLES:
sage: G.<a,b> = FreeGroup('a, b') sage: x = G([1, 2, -1, -2]) sage: x.is_one() False sage: (x * ~x).is_one() True
- multiplicative_order()#
Return the multiplicative order.
EXAMPLES:
sage: from sage.groups.libgap_group import GroupLibGAP sage: G = GroupLibGAP(libgap.GL(2, 3)) sage: a,b = G.gens() sage: print(a.order()) 2 sage: print(a.multiplicative_order()) 2 sage: z = Mod(0, 3) sage: o = Mod(1, 3) sage: G(libgap([[o,o],[z,o]])).order() 3
- normalizer()#
Return the normalizer of the cyclic group generated by this element.
EXAMPLES:
sage: from sage.groups.libgap_group import GroupLibGAP sage: G = GroupLibGAP(libgap.GL(3,3)) sage: a,b = G.gens() sage: H = a.normalizer() sage: H <group of 3x3 matrices over GF(3)> sage: H.cardinality() 96 sage: all(g*a == a*g for g in H) True
- nth_roots(n)#
Return the set of
n
-th roots of this group element.EXAMPLES:
sage: from sage.groups.libgap_group import GroupLibGAP sage: G = GroupLibGAP(libgap.GL(3,3)) sage: a,b = G.gens() sage: g = a*b**2*a*~b sage: r = g.nth_roots(4) sage: r [[ [ Z(3), Z(3), Z(3)^0 ], [ Z(3)^0, Z(3)^0, 0*Z(3) ], [ 0*Z(3), Z(3), 0*Z(3) ] ], [ [ Z(3)^0, Z(3)^0, Z(3) ], [ Z(3), Z(3), 0*Z(3) ], [ 0*Z(3), Z(3)^0, 0*Z(3) ] ]] sage: r[0]**4 == r[1]**4 == g True
- order()#
Return the multiplicative order.
EXAMPLES:
sage: from sage.groups.libgap_group import GroupLibGAP sage: G = GroupLibGAP(libgap.GL(2, 3)) sage: a,b = G.gens() sage: print(a.order()) 2 sage: print(a.multiplicative_order()) 2 sage: z = Mod(0, 3) sage: o = Mod(1, 3) sage: G(libgap([[o,o],[z,o]])).order() 3
- class sage.groups.libgap_wrapper.ParentLibGAP(libgap_parent, ambient=None)#
Bases:
SageObject
A class for parents to keep track of the GAP parent.
This is not a complete group in Sage, this class is only a base class that you can use to implement your own groups with LibGAP. See
libgap_group
for a minimal example of a group that is actually usable.Your implementation definitely needs to supply
__reduce__()
: serialize the LibGAP group. Since GAP does not support Python pickles natively, you need to figure out yourself how you can recreate the group from a pickle.
INPUT:
libgap_parent
– the libgap element that is the parent in GAP.ambient
– A derived class ofParentLibGAP
orNone
(default). The ambient class iflibgap_parent
has been defined as a subgroup.
EXAMPLES:
sage: from sage.groups.libgap_wrapper import ElementLibGAP, ParentLibGAP sage: from sage.groups.group import Group sage: class FooElement(ElementLibGAP): ....: pass sage: class FooGroup(Group, ParentLibGAP): ....: Element = FooElement ....: def __init__(self): ....: lg = libgap(libgap.CyclicGroup(3)) # dummy ....: ParentLibGAP.__init__(self, lg) ....: Group.__init__(self) sage: FooGroup() <pc group of size 3 with 1 generators>
- ambient()#
Return the ambient group of a subgroup.
OUTPUT:
A group containing
self
. Ifself
has not been defined as a subgroup, we just returnself
.EXAMPLES:
sage: G = FreeGroup(3) sage: G.ambient() is G True
- gap()#
Return the gap representation of self.
OUTPUT:
EXAMPLES:
sage: G = FreeGroup(3); G Free Group on generators {x0, x1, x2} sage: G.gap() <free group on the generators [ x0, x1, x2 ]> sage: G.gap().parent() C library interface to GAP sage: type(G.gap()) <class 'sage.libs.gap.element.GapElement'>
This can be useful, for example, to call GAP functions that are not wrapped in Sage:
sage: G = FreeGroup(3) sage: H = G.gap() sage: H.DirectProduct(H) <fp group on the generators [ f1, f2, f3, f4, f5, f6 ]> sage: H.DirectProduct(H).RelatorsOfFpGroup() [ f1^-1*f4^-1*f1*f4, f1^-1*f5^-1*f1*f5, f1^-1*f6^-1*f1*f6, f2^-1*f4^-1*f2*f4, f2^-1*f5^-1*f2*f5, f2^-1*f6^-1*f2*f6, f3^-1*f4^-1*f3*f4, f3^-1*f5^-1*f3*f5, f3^-1*f6^-1*f3*f6 ]
We can also convert directly to libgap:
sage: libgap(GL(2, ZZ)) GL(2,Integers)
- gen(i)#
Return the \(i\)-th generator of self.
Warning
Indexing starts at \(0\) as usual in Sage/Python. Not as in GAP, where indexing starts at \(1\).
INPUT:
i
– integer between \(0\) (inclusive) andngens()
(exclusive). The index of the generator.
OUTPUT:
The \(i\)-th generator of the group.
EXAMPLES:
sage: G = FreeGroup('a, b') sage: G.gen(0) a sage: G.gen(1) b
- generators()#
Return the generators of the group.
EXAMPLES:
sage: G = FreeGroup(2) sage: G.gens() (x0, x1) sage: H = FreeGroup('a, b, c') sage: H.gens() (a, b, c)
generators()
is an alias forgens()
sage: G = FreeGroup('a, b') sage: G.generators() (a, b) sage: H = FreeGroup(3, 'x') sage: H.generators() (x0, x1, x2)
- gens()#
Return the generators of the group.
EXAMPLES:
sage: G = FreeGroup(2) sage: G.gens() (x0, x1) sage: H = FreeGroup('a, b, c') sage: H.gens() (a, b, c)
generators()
is an alias forgens()
sage: G = FreeGroup('a, b') sage: G.generators() (a, b) sage: H = FreeGroup(3, 'x') sage: H.generators() (x0, x1, x2)
- is_subgroup()#
Return whether the group was defined as a subgroup of a bigger group.
You can access the containing group with
ambient()
.OUTPUT:
Boolean.
EXAMPLES:
sage: G = FreeGroup(3) sage: G.is_subgroup() False
- ngens()#
Return the number of generators of self.
OUTPUT:
Integer.
EXAMPLES:
sage: G = FreeGroup(2) sage: G.ngens() 2
- one()#
Return the identity element of self.
EXAMPLES:
sage: G = FreeGroup(3) sage: G.one() 1 sage: G.one() == G([]) True sage: G.one().Tietze() ()
- subgroup(generators)#
Return the subgroup generated.
INPUT:
generators
– a list/tuple/iterable of group elements.
OUTPUT:
The subgroup generated by
generators
.EXAMPLES:
sage: F.<a,b> = FreeGroup() sage: G = F.subgroup([a^2*b]); G Group([ a^2*b ]) sage: G.gens() (a^2*b,)
We check that coercions between the subgroup and its ambient group work:
sage: F.0 * G.0 a^3*b
Checking that trac ticket #19270 is fixed:
sage: gens = [w.matrix() for w in WeylGroup(['B', 3])] sage: G = MatrixGroup(gens) sage: import itertools sage: diagonals = itertools.product((1,-1), repeat=3) sage: subgroup_gens = [diagonal_matrix(L) for L in diagonals] sage: G.subgroup(subgroup_gens) Subgroup with 8 generators of Matrix group over Rational Field with 48 generators