GAP element wrapper¶

This document describes the individual wrappers for various GAP elements. For general information about GAP, you should read the libgap module documentation.

class sage.libs.gap.element.GapElement[source]¶

Bases: RingElement

Wrapper for all Gap objects.

Note

In order to create GapElements you should use the libgap instance (the parent of all Gap elements) to convert things into GapElement. You must not create GapElement instances manually.

EXAMPLES:

Sage

sage: libgap(0)
0

Python

>>> from sage.all import *
>>> libgap(Integer(0))
0

If Gap finds an error while evaluating, a GAPError exception is raised:

Sage

sage: libgap.eval('1/0')
Traceback (most recent call last):
...
GAPError: Error, Rational operations: <divisor> must not be zero

Python

>>> from sage.all import *
>>> libgap.eval('1/0')
Traceback (most recent call last):
...
GAPError: Error, Rational operations: <divisor> must not be zero

Also, a GAPError is raised if the input is not a simple expression:

Sage

sage: libgap.eval('1; 2; 3')
Traceback (most recent call last):
...
GAPError: can only evaluate a single statement

Python

>>> from sage.all import *
>>> libgap.eval('1; 2; 3')
Traceback (most recent call last):
...
GAPError: can only evaluate a single statement

deepcopy(mut)[source]¶

Return a deepcopy of this Gap object.

Note that this is the same thing as calling StructuralCopy but much faster.

INPUT:

mut – boolean; whether to return a mutable copy

EXAMPLES:

Sage

sage: a = libgap([[0,1],[2,3]])
sage: b = a.deepcopy(1)
sage: b[0,0] = 5
sage: a
[ [ 0, 1 ], [ 2, 3 ] ]
sage: b
[ [ 5, 1 ], [ 2, 3 ] ]

sage: l = libgap([0,1])
sage: l.deepcopy(0).IsMutable()
false
sage: l.deepcopy(1).IsMutable()
true

Python

>>> from sage.all import *
>>> a = libgap([[Integer(0),Integer(1)],[Integer(2),Integer(3)]])
>>> b = a.deepcopy(Integer(1))
>>> b[Integer(0),Integer(0)] = Integer(5)
>>> a
[ [ 0, 1 ], [ 2, 3 ] ]
>>> b
[ [ 5, 1 ], [ 2, 3 ] ]

>>> l = libgap([Integer(0),Integer(1)])
>>> l.deepcopy(Integer(0)).IsMutable()
false
>>> l.deepcopy(Integer(1)).IsMutable()
true

is_bool()[source]¶

Return whether the wrapped GAP object is a GAP boolean.

OUTPUT: boolean

EXAMPLES:

Sage

sage: libgap(True).is_bool()
True

Python

>>> from sage.all import *
>>> libgap(True).is_bool()
True

is_function()[source]¶

Return whether the wrapped GAP object is a function.

OUTPUT: boolean

EXAMPLES:

Sage

sage: a = libgap.eval("NormalSubgroups")
sage: a.is_function()
True
sage: a = libgap(2/3)
sage: a.is_function()
False

Python

>>> from sage.all import *
>>> a = libgap.eval("NormalSubgroups")
>>> a.is_function()
True
>>> a = libgap(Integer(2)/Integer(3))
>>> a.is_function()
False

is_list()[source]¶

Return whether the wrapped GAP object is a GAP List.

OUTPUT: boolean

EXAMPLES:

Sage

sage: libgap.eval('[1, 2,,,, 5]').is_list()
True
sage: libgap.eval('3/2').is_list()
False

Python

>>> from sage.all import *
>>> libgap.eval('[1, 2,,,, 5]').is_list()
True
>>> libgap.eval('3/2').is_list()
False

is_permutation()[source]¶

Return whether the wrapped GAP object is a GAP permutation.

OUTPUT: boolean

EXAMPLES:

Sage

sage: perm = libgap.PermList( libgap([1,5,2,3,4]) );  perm
(2,5,4,3)
sage: perm.is_permutation()
True
sage: libgap('this is a string').is_permutation()
False

Python

>>> from sage.all import *
>>> perm = libgap.PermList( libgap([Integer(1),Integer(5),Integer(2),Integer(3),Integer(4)]) );  perm
(2,5,4,3)
>>> perm.is_permutation()
True
>>> libgap('this is a string').is_permutation()
False

is_record()[source]¶

Return whether the wrapped GAP object is a GAP record.

OUTPUT: boolean

EXAMPLES:

Sage

sage: libgap.eval('[1, 2,,,, 5]').is_record()
False
sage: libgap.eval('rec(a:=1, b:=3)').is_record()
True

Python

>>> from sage.all import *
>>> libgap.eval('[1, 2,,,, 5]').is_record()
False
>>> libgap.eval('rec(a:=1, b:=3)').is_record()
True

is_string()[source]¶

Return whether the wrapped GAP object is a GAP string.

OUTPUT: boolean

EXAMPLES:

Sage

sage: libgap('this is a string').is_string()
True

Python

>>> from sage.all import *
>>> libgap('this is a string').is_string()
True

sage()[source]¶

Return the Sage equivalent of the GapElement.

EXAMPLES:

Sage

sage: libgap(1).sage()
1
sage: type(_)
<class 'sage.rings.integer.Integer'>

sage: libgap(3/7).sage()
3/7
sage: type(_)
<class 'sage.rings.rational.Rational'>

sage: libgap.eval('5 + 7*E(3)').sage()
7*zeta3 + 5

sage: libgap(Infinity).sage()
+Infinity
sage: libgap(-Infinity).sage()
-Infinity

sage: libgap(True).sage()
True
sage: libgap(False).sage()
False
sage: type(_)
<... 'bool'>

sage: libgap('this is a string').sage()
'this is a string'
sage: type(_)
<... 'str'>

sage: x = libgap.Integers.Indeterminate("x")

sage: p = x^2 - 2*x + 3
sage: p.sage()
x^2 - 2*x + 3
sage: p.sage().parent()
Univariate Polynomial Ring in x over Integer Ring

sage: p = x^-2 + 3*x
sage: p.sage()
x^-2 + 3*x
sage: p.sage().parent()
Univariate Laurent Polynomial Ring in x over Integer Ring

sage: p = (3 * x^2 + x) / (x^2 - 2)
sage: p.sage()
(3*x^2 + x)/(x^2 - 2)
sage: p.sage().parent()
Fraction Field of Univariate Polynomial Ring in x over Integer Ring

sage: G0 = libgap.SymplecticGroup(4,2)
sage: P = G0.IsomorphismFpGroup().Range()
sage: G = P.sage()
sage: G.gap() == P
True

sage: F0 = libgap.FreeGroup(2)
sage: F = F0.sage()
sage: F.gap() is F0
True

Python

>>> from sage.all import *
>>> libgap(Integer(1)).sage()
1
>>> type(_)
<class 'sage.rings.integer.Integer'>

>>> libgap(Integer(3)/Integer(7)).sage()
3/7
>>> type(_)
<class 'sage.rings.rational.Rational'>

>>> libgap.eval('5 + 7*E(3)').sage()
7*zeta3 + 5

>>> libgap(Infinity).sage()
+Infinity
>>> libgap(-Infinity).sage()
-Infinity

>>> libgap(True).sage()
True
>>> libgap(False).sage()
False
>>> type(_)
<... 'bool'>

>>> libgap('this is a string').sage()
'this is a string'
>>> type(_)
<... 'str'>

>>> x = libgap.Integers.Indeterminate("x")

>>> p = x**Integer(2) - Integer(2)*x + Integer(3)
>>> p.sage()
x^2 - 2*x + 3
>>> p.sage().parent()
Univariate Polynomial Ring in x over Integer Ring

>>> p = x**-Integer(2) + Integer(3)*x
>>> p.sage()
x^-2 + 3*x
>>> p.sage().parent()
Univariate Laurent Polynomial Ring in x over Integer Ring

>>> p = (Integer(3) * x**Integer(2) + x) / (x**Integer(2) - Integer(2))
>>> p.sage()
(3*x^2 + x)/(x^2 - 2)
>>> p.sage().parent()
Fraction Field of Univariate Polynomial Ring in x over Integer Ring

>>> G0 = libgap.SymplecticGroup(Integer(4),Integer(2))
>>> P = G0.IsomorphismFpGroup().Range()
>>> G = P.sage()
>>> G.gap() == P
True

>>> F0 = libgap.FreeGroup(Integer(2))
>>> F = F0.sage()
>>> F.gap() is F0
True

class sage.libs.gap.element.GapElement_Boolean[source]¶

Bases: GapElement

Derived class of GapElement for GAP boolean values.

EXAMPLES:

Sage

sage: b = libgap(True)
sage: type(b)
<class 'sage.libs.gap.element.GapElement_Boolean'>

Python

>>> from sage.all import *
>>> b = libgap(True)
>>> type(b)
<class 'sage.libs.gap.element.GapElement_Boolean'>

sage()[source]¶

Return the Sage equivalent of the GapElement.

OUTPUT:

A Python boolean if the values is either true or false. GAP booleans can have the third value Fail, in which case a ValueError is raised.

EXAMPLES:

Sage

sage: b = libgap.eval('true');  b
true
sage: type(_)
<class 'sage.libs.gap.element.GapElement_Boolean'>
sage: b.sage()
True
sage: type(_)
<... 'bool'>

sage: libgap.eval('fail')
fail
sage: _.sage()
Traceback (most recent call last):
...
ValueError: the GAP boolean value "fail" cannot be represented in Sage

Python

>>> from sage.all import *
>>> b = libgap.eval('true');  b
true
>>> type(_)
<class 'sage.libs.gap.element.GapElement_Boolean'>
>>> b.sage()
True
>>> type(_)
<... 'bool'>

>>> libgap.eval('fail')
fail
>>> _.sage()
Traceback (most recent call last):
...
ValueError: the GAP boolean value "fail" cannot be represented in Sage

class sage.libs.gap.element.GapElement_Cyclotomic[source]¶

Bases: GapElement

Derived class of GapElement for GAP universal cyclotomics.

EXAMPLES:

Sage

sage: libgap.eval('E(3)')
E(3)
sage: type(_)
<class 'sage.libs.gap.element.GapElement_Cyclotomic'>

Python

>>> from sage.all import *
>>> libgap.eval('E(3)')
E(3)
>>> type(_)
<class 'sage.libs.gap.element.GapElement_Cyclotomic'>

sage(ring=None)[source]¶

Return the Sage equivalent of the GapElement_Cyclotomic.

INPUT:

ring – a Sage cyclotomic field or None (default); if not specified, a suitable minimal cyclotomic field will be constructed

OUTPUT: a Sage cyclotomic field element

EXAMPLES:

Sage

sage: n = libgap.eval('E(3)')
sage: n.sage()
zeta3
sage: parent(_)
Cyclotomic Field of order 3 and degree 2

sage: n.sage(ring=CyclotomicField(6))
zeta6 - 1

sage: libgap.E(3).sage(ring=CyclotomicField(3))
zeta3
sage: libgap.E(3).sage(ring=CyclotomicField(6))
zeta6 - 1

Python

>>> from sage.all import *
>>> n = libgap.eval('E(3)')
>>> n.sage()
zeta3
>>> parent(_)
Cyclotomic Field of order 3 and degree 2

>>> n.sage(ring=CyclotomicField(Integer(6)))
zeta6 - 1

>>> libgap.E(Integer(3)).sage(ring=CyclotomicField(Integer(3)))
zeta3
>>> libgap.E(Integer(3)).sage(ring=CyclotomicField(Integer(6)))
zeta6 - 1

class sage.libs.gap.element.GapElement_FiniteField[source]¶

Bases: GapElement

Derived class of GapElement for GAP finite field elements.

EXAMPLES:

Sage

sage: libgap.eval('Z(5)^2')
Z(5)^2
sage: type(_)
<class 'sage.libs.gap.element.GapElement_FiniteField'>

Python

>>> from sage.all import *
>>> libgap.eval('Z(5)^2')
Z(5)^2
>>> type(_)
<class 'sage.libs.gap.element.GapElement_FiniteField'>

lift()[source]¶

Return an integer lift.

OUTPUT:

The smallest positive GapElement_Integer that equals self in the prime finite field.

EXAMPLES:

Sage

sage: n = libgap.eval('Z(5)^2')
sage: n.lift()
4
sage: type(_)
<class 'sage.libs.gap.element.GapElement_Integer'>

sage: n = libgap.eval('Z(25)')
sage: n.lift()
Traceback (most recent call last):
TypeError: not in prime subfield

Python

>>> from sage.all import *
>>> n = libgap.eval('Z(5)^2')
>>> n.lift()
4
>>> type(_)
<class 'sage.libs.gap.element.GapElement_Integer'>

>>> n = libgap.eval('Z(25)')
>>> n.lift()
Traceback (most recent call last):
TypeError: not in prime subfield

sage(ring=None, var='a')[source]¶

Return the Sage equivalent of the GapElement_FiniteField.

INPUT:

ring – a Sage finite field or None (default). The field to return self in. If not specified, a suitable finite field will be constructed.

OUTPUT:

A Sage finite field element. The isomorphism is chosen such that the Gap PrimitiveRoot() maps to the Sage multiplicative_generator().

EXAMPLES:

Sage

sage: n = libgap.eval('Z(25)^2')
sage: n.sage()
a + 3
sage: parent(_)
Finite Field in a of size 5^2

sage: n.sage(ring=GF(5))
Traceback (most recent call last):
...
ValueError: the given ring is incompatible ...

Python

>>> from sage.all import *
>>> n = libgap.eval('Z(25)^2')
>>> n.sage()
a + 3
>>> parent(_)
Finite Field in a of size 5^2

>>> n.sage(ring=GF(Integer(5)))
Traceback (most recent call last):
...
ValueError: the given ring is incompatible ...

class sage.libs.gap.element.GapElement_Float[source]¶

Bases: GapElement

Derived class of GapElement for GAP floating point numbers.

EXAMPLES:

Sage

sage: i = libgap(123.5)
sage: type(i)
<class 'sage.libs.gap.element.GapElement_Float'>
sage: RDF(i)
123.5
sage: float(i)
123.5

Python

>>> from sage.all import *
>>> i = libgap(RealNumber('123.5'))
>>> type(i)
<class 'sage.libs.gap.element.GapElement_Float'>
>>> RDF(i)
123.5
>>> float(i)
123.5

sage(ring=None)[source]¶

Return the Sage equivalent of the GapElement_Float.

ring – a floating point field or None (default); if not specified, the default Sage RDF is used

OUTPUT: a Sage double precision floating point number

EXAMPLES:

Sage

sage: a = libgap.eval("Float(3.25)").sage()
sage: a
3.25
sage: parent(a)
Real Double Field

Python

>>> from sage.all import *
>>> a = libgap.eval("Float(3.25)").sage()
>>> a
3.25
>>> parent(a)
Real Double Field

class sage.libs.gap.element.GapElement_Function[source]¶

Bases: GapElement

Derived class of GapElement for GAP functions.

EXAMPLES:

Sage

sage: f = libgap.Cycles
sage: type(f)
<class 'sage.libs.gap.element.GapElement_Function'>

Python

>>> from sage.all import *
>>> f = libgap.Cycles
>>> type(f)
<class 'sage.libs.gap.element.GapElement_Function'>

class sage.libs.gap.element.GapElement_Integer[source]¶

Bases: GapElement

Derived class of GapElement for GAP integers.

EXAMPLES:

Sage

sage: i = libgap(123)
sage: type(i)
<class 'sage.libs.gap.element.GapElement_Integer'>
sage: ZZ(i)
123

Python

>>> from sage.all import *
>>> i = libgap(Integer(123))
>>> type(i)
<class 'sage.libs.gap.element.GapElement_Integer'>
>>> ZZ(i)
123

is_C_int()[source]¶

Return whether the wrapped GAP object is a immediate GAP integer.

An immediate integer is one that is stored as a C integer, and is subject to the usual size limits. Larger integers are stored in GAP as GMP integers.

OUTPUT: boolean

EXAMPLES:

Sage

sage: n = libgap(1)
sage: type(n)
<class 'sage.libs.gap.element.GapElement_Integer'>
sage: n.is_C_int()
True
sage: n.IsInt()
true

sage: N = libgap(2^130)
sage: type(N)
<class 'sage.libs.gap.element.GapElement_Integer'>
sage: N.is_C_int()
False
sage: N.IsInt()
true

Python

>>> from sage.all import *
>>> n = libgap(Integer(1))
>>> type(n)
<class 'sage.libs.gap.element.GapElement_Integer'>
>>> n.is_C_int()
True
>>> n.IsInt()
true

>>> N = libgap(Integer(2)**Integer(130))
>>> type(N)
<class 'sage.libs.gap.element.GapElement_Integer'>
>>> N.is_C_int()
False
>>> N.IsInt()
true

sage(ring=None)[source]¶

Return the Sage equivalent of the GapElement_Integer.

ring – integer ring or None (default); if not specified, the default Sage integer ring is used

OUTPUT: a Sage integer

EXAMPLES:

Sage

sage: libgap([ 1, 3, 4 ]).sage()
[1, 3, 4]
sage: all( x in ZZ for x in _ )
True

sage: libgap(132).sage(ring=IntegerModRing(13))
2
sage: parent(_)
Ring of integers modulo 13

Python

>>> from sage.all import *
>>> libgap([ Integer(1), Integer(3), Integer(4) ]).sage()
[1, 3, 4]
>>> all( x in ZZ for x in _ )
True

>>> libgap(Integer(132)).sage(ring=IntegerModRing(Integer(13)))
2
>>> parent(_)
Ring of integers modulo 13

class sage.libs.gap.element.GapElement_IntegerMod[source]¶

Bases: GapElement

Derived class of GapElement for GAP integers modulo an integer.

EXAMPLES:

Sage

sage: n = IntegerModRing(123)(13)
sage: i = libgap(n)
sage: type(i)
<class 'sage.libs.gap.element.GapElement_IntegerMod'>

Python

>>> from sage.all import *
>>> n = IntegerModRing(Integer(123))(Integer(13))
>>> i = libgap(n)
>>> type(i)
<class 'sage.libs.gap.element.GapElement_IntegerMod'>

lift()[source]¶

Return an integer lift.

OUTPUT:

A GapElement_Integer that equals self in the integer mod ring.

EXAMPLES:

Sage

sage: n = libgap.eval('One(ZmodnZ(123)) * 13')
sage: n.lift()
13
sage: type(_)
<class 'sage.libs.gap.element.GapElement_Integer'>

Python

>>> from sage.all import *
>>> n = libgap.eval('One(ZmodnZ(123)) * 13')
>>> n.lift()
13
>>> type(_)
<class 'sage.libs.gap.element.GapElement_Integer'>

sage(ring=None)[source]¶

Return the Sage equivalent of the GapElement_IntegerMod.

INPUT:

ring – Sage integer mod ring or None (default); if not specified, a suitable integer mod ring is used automatically

OUTPUT: a Sage integer modulo another integer

EXAMPLES:

Sage

sage: n = libgap.eval('One(ZmodnZ(123)) * 13')
sage: n.sage()
13
sage: parent(_)
Ring of integers modulo 123

Python

>>> from sage.all import *
>>> n = libgap.eval('One(ZmodnZ(123)) * 13')
>>> n.sage()
13
>>> parent(_)
Ring of integers modulo 123

class sage.libs.gap.element.GapElement_List[source]¶

Bases: GapElement

Derived class of GapElement for GAP Lists.

Note

Lists are indexed by \(0..len(l)-1\), as expected from Python. This differs from the GAP convention where lists start at \(1\).

EXAMPLES:

Sage

sage: lst = libgap.SymmetricGroup(3).List(); lst
[ (), (1,3), (1,2,3), (2,3), (1,3,2), (1,2) ]
sage: type(lst)
<class 'sage.libs.gap.element.GapElement_List'>
sage: len(lst)
6
sage: lst[3]
(2,3)

Python

>>> from sage.all import *
>>> lst = libgap.SymmetricGroup(Integer(3)).List(); lst
[ (), (1,3), (1,2,3), (2,3), (1,3,2), (1,2) ]
>>> type(lst)
<class 'sage.libs.gap.element.GapElement_List'>
>>> len(lst)
6
>>> lst[Integer(3)]
(2,3)

We can easily convert a Gap List object into a Python list:

Sage

sage: list(lst)
[(), (1,3), (1,2,3), (2,3), (1,3,2), (1,2)]
sage: type(_)
<... 'list'>

Python

>>> from sage.all import *
>>> list(lst)
[(), (1,3), (1,2,3), (2,3), (1,3,2), (1,2)]
>>> type(_)
<... 'list'>

Range checking is performed:

Sage

sage: lst[10]
Traceback (most recent call last):
...
IndexError: index out of range.

Python

>>> from sage.all import *
>>> lst[Integer(10)]
Traceback (most recent call last):
...
IndexError: index out of range.

matrix(ring=None)[source]¶

Return the list as a matrix.

GAP does not have a special matrix data type, they are just lists of lists. This function converts a GAP list of lists to a Sage matrix.

OUTPUT: a Sage matrix

EXAMPLES:

Sage

sage: F = libgap.GF(4)
sage: a = F.PrimitiveElement()
sage: m = libgap([[a,a^0],[0*a,a^2]]); m
[ [ Z(2^2), Z(2)^0 ],
  [ 0*Z(2), Z(2^2)^2 ] ]
sage: m.IsMatrix()
true
sage: matrix(m)
[    a     1]
[    0 a + 1]
sage: matrix(GF(4,'B'), m)
[    B     1]
[    0 B + 1]

sage: M = libgap.eval('SL(2,GF(5))').GeneratorsOfGroup()[1]
sage: type(M)
<class 'sage.libs.gap.element.GapElement_List'>
sage: M[0][0]
Z(5)^2
sage: M.IsMatrix()
true
sage: M.matrix()
[4 1]
[4 0]

Python

>>> from sage.all import *
>>> F = libgap.GF(Integer(4))
>>> a = F.PrimitiveElement()
>>> m = libgap([[a,a**Integer(0)],[Integer(0)*a,a**Integer(2)]]); m
[ [ Z(2^2), Z(2)^0 ],
  [ 0*Z(2), Z(2^2)^2 ] ]
>>> m.IsMatrix()
true
>>> matrix(m)
[    a     1]
[    0 a + 1]
>>> matrix(GF(Integer(4),'B'), m)
[    B     1]
[    0 B + 1]

>>> M = libgap.eval('SL(2,GF(5))').GeneratorsOfGroup()[Integer(1)]
>>> type(M)
<class 'sage.libs.gap.element.GapElement_List'>
>>> M[Integer(0)][Integer(0)]
Z(5)^2
>>> M.IsMatrix()
true
>>> M.matrix()
[4 1]
[4 0]

sage(**kwds)[source]¶

Return the Sage equivalent of the GapElement.

OUTPUT: a Python list

EXAMPLES:

Sage

sage: libgap([ 1, 3, 4 ]).sage()
[1, 3, 4]
sage: all( x in ZZ for x in _ )
True

Python

>>> from sage.all import *
>>> libgap([ Integer(1), Integer(3), Integer(4) ]).sage()
[1, 3, 4]
>>> all( x in ZZ for x in _ )
True

vector(ring=None)[source]¶

Return the list as a vector.

GAP does not have a special vector data type, they are just lists. This function converts a GAP list to a Sage vector.

OUTPUT: a Sage vector

EXAMPLES:

Sage

sage: F = libgap.GF(4)
sage: a = F.PrimitiveElement()
sage: m = libgap([0*a, a, a^3, a^2]); m
[ 0*Z(2), Z(2^2), Z(2)^0, Z(2^2)^2 ]
sage: type(m)
<class 'sage.libs.gap.element.GapElement_List'>
sage: m[3]
Z(2^2)^2
sage: vector(m)
(0, a, 1, a + 1)
sage: vector(GF(4,'B'), m)
(0, B, 1, B + 1)

Python

>>> from sage.all import *
>>> F = libgap.GF(Integer(4))
>>> a = F.PrimitiveElement()
>>> m = libgap([Integer(0)*a, a, a**Integer(3), a**Integer(2)]); m
[ 0*Z(2), Z(2^2), Z(2)^0, Z(2^2)^2 ]
>>> type(m)
<class 'sage.libs.gap.element.GapElement_List'>
>>> m[Integer(3)]
Z(2^2)^2
>>> vector(m)
(0, a, 1, a + 1)
>>> vector(GF(Integer(4),'B'), m)
(0, B, 1, B + 1)

class sage.libs.gap.element.GapElement_MethodProxy[source]¶

Bases: GapElement_Function

Helper class returned by GapElement.__getattr__.

Derived class of GapElement for GAP functions. Like its parent, you can call instances to implement function call syntax. The only difference is that a fixed first argument is prepended to the argument list.

EXAMPLES:

Sage

sage: lst = libgap([])
sage: lst.Add
<Gap function "Add">
sage: type(_)
<class 'sage.libs.gap.element.GapElement_MethodProxy'>
sage: lst.Add(1)
sage: lst
[ 1 ]

Python

>>> from sage.all import *
>>> lst = libgap([])
>>> lst.Add
<Gap function "Add">
>>> type(_)
<class 'sage.libs.gap.element.GapElement_MethodProxy'>
>>> lst.Add(Integer(1))
>>> lst
[ 1 ]

class sage.libs.gap.element.GapElement_Permutation[source]¶

Bases: GapElement

Derived class of GapElement for GAP permutations.

Note

Permutations in GAP act on the numbers starting with 1.

EXAMPLES:

Sage

sage: perm = libgap.eval('(1,5,2)(4,3,8)')
sage: type(perm)
<class 'sage.libs.gap.element.GapElement_Permutation'>

Python

>>> from sage.all import *
>>> perm = libgap.eval('(1,5,2)(4,3,8)')
>>> type(perm)
<class 'sage.libs.gap.element.GapElement_Permutation'>

sage(parent=None)[source]¶

Return the Sage equivalent of the GapElement.

If the permutation group is given as parent, this method is much faster.

EXAMPLES:

Sage

sage: perm_gap = libgap.eval('(1,5,2)(4,3,8)');  perm_gap
(1,5,2)(3,8,4)
sage: perm_gap.sage()
[5, 1, 8, 3, 2, 6, 7, 4]
sage: type(_)
<class 'sage.combinat.permutation.StandardPermutations_all_with_category.element_class'>
sage: perm_gap.sage(PermutationGroup([(1,2),(1,2,3,4,5,6,7,8)]))
(1,5,2)(3,8,4)
sage: type(_)
<class 'sage.groups.perm_gps.permgroup_element.PermutationGroupElement'>

Python

>>> from sage.all import *
>>> perm_gap = libgap.eval('(1,5,2)(4,3,8)');  perm_gap
(1,5,2)(3,8,4)
>>> perm_gap.sage()
[5, 1, 8, 3, 2, 6, 7, 4]
>>> type(_)
<class 'sage.combinat.permutation.StandardPermutations_all_with_category.element_class'>
>>> perm_gap.sage(PermutationGroup([(Integer(1),Integer(2)),(Integer(1),Integer(2),Integer(3),Integer(4),Integer(5),Integer(6),Integer(7),Integer(8))]))
(1,5,2)(3,8,4)
>>> type(_)
<class 'sage.groups.perm_gps.permgroup_element.PermutationGroupElement'>

class sage.libs.gap.element.GapElement_Rational[source]¶

Bases: GapElement

Derived class of GapElement for GAP rational numbers.

EXAMPLES:

Sage

sage: r = libgap(123/456)
sage: type(r)
<class 'sage.libs.gap.element.GapElement_Rational'>

Python

>>> from sage.all import *
>>> r = libgap(Integer(123)/Integer(456))
>>> type(r)
<class 'sage.libs.gap.element.GapElement_Rational'>

sage(ring=None)[source]¶

Return the Sage equivalent of the GapElement.

INPUT:

ring – the Sage rational ring or None (default); if not specified, the rational ring is used automatically

OUTPUT: a Sage rational number

EXAMPLES:

Sage

sage: r = libgap(123/456);  r
41/152
sage: type(_)
<class 'sage.libs.gap.element.GapElement_Rational'>
sage: r.sage()
41/152
sage: type(_)
<class 'sage.rings.rational.Rational'>

Python

>>> from sage.all import *
>>> r = libgap(Integer(123)/Integer(456));  r
41/152
>>> type(_)
<class 'sage.libs.gap.element.GapElement_Rational'>
>>> r.sage()
41/152
>>> type(_)
<class 'sage.rings.rational.Rational'>

class sage.libs.gap.element.GapElement_Record[source]¶

Bases: GapElement

Derived class of GapElement for GAP records.

EXAMPLES:

Sage

sage: rec = libgap.eval('rec(a:=123, b:=456)')
sage: type(rec)
<class 'sage.libs.gap.element.GapElement_Record'>
sage: len(rec)
2
sage: rec['a']
123

Python

>>> from sage.all import *
>>> rec = libgap.eval('rec(a:=123, b:=456)')
>>> type(rec)
<class 'sage.libs.gap.element.GapElement_Record'>
>>> len(rec)
2
>>> rec['a']
123

We can easily convert a Gap rec object into a Python dict:

Sage

sage: dict(rec)
{'a': 123, 'b': 456}
sage: type(_)
<... 'dict'>

Python

>>> from sage.all import *
>>> dict(rec)
{'a': 123, 'b': 456}
>>> type(_)
<... 'dict'>

Range checking is performed:

Sage

sage: rec['no_such_element']
Traceback (most recent call last):
...
GAPError: Error, Record Element: '<rec>.no_such_element' must have an assigned value

Python

>>> from sage.all import *
>>> rec['no_such_element']
Traceback (most recent call last):
...
GAPError: Error, Record Element: '<rec>.no_such_element' must have an assigned value

record_name_to_index(name)[source]¶

Convert string to GAP record index.

INPUT:

py_name – a python string

OUTPUT:

A UInt, which is a GAP hash of the string. If this is the first time the string is encountered, a new integer is returned(!)

EXAMPLES:

Sage

sage: rec = libgap.eval('rec(first:=123, second:=456)')
sage: rec.record_name_to_index('first')   # random output
1812
sage: rec.record_name_to_index('no_such_name') # random output
3776

Python

>>> from sage.all import *
>>> rec = libgap.eval('rec(first:=123, second:=456)')
>>> rec.record_name_to_index('first')   # random output
1812
>>> rec.record_name_to_index('no_such_name') # random output
3776

sage()[source]¶

Return the Sage equivalent of the GapElement.

EXAMPLES:

Sage

sage: libgap.eval('rec(a:=1, b:=2)').sage()
{'a': 1, 'b': 2}
sage: all( isinstance(key,str) and val in ZZ for key,val in _.items() )
True

sage: rec = libgap.eval('rec(a:=123, b:=456, Sym3:=SymmetricGroup(3))')
sage: rec.sage()
{'Sym3': NotImplementedError('cannot construct equivalent Sage object'...),
 'a': 123,
 'b': 456}

Python

>>> from sage.all import *
>>> libgap.eval('rec(a:=1, b:=2)').sage()
{'a': 1, 'b': 2}
>>> all( isinstance(key,str) and val in ZZ for key,val in _.items() )
True

>>> rec = libgap.eval('rec(a:=123, b:=456, Sym3:=SymmetricGroup(3))')
>>> rec.sage()
{'Sym3': NotImplementedError('cannot construct equivalent Sage object'...),
 'a': 123,
 'b': 456}

class sage.libs.gap.element.GapElement_RecordIterator[source]¶

Bases: object

Iterator for GapElement_Record.

Since Cython does not support generators yet, we implement the older iterator specification with this auxiliary class.

INPUT:

rec – the GapElement_Record to iterate over

EXAMPLES:

Sage

sage: rec = libgap.eval('rec(a:=123, b:=456)')
sage: sorted(rec)
[('a', 123), ('b', 456)]
sage: dict(rec)
{'a': 123, 'b': 456}

Python

>>> from sage.all import *
>>> rec = libgap.eval('rec(a:=123, b:=456)')
>>> sorted(rec)
[('a', 123), ('b', 456)]
>>> dict(rec)
{'a': 123, 'b': 456}

class sage.libs.gap.element.GapElement_Ring[source]¶

Bases: GapElement

Derived class of GapElement for GAP rings (parents of ring elements).

EXAMPLES:

Sage

sage: i = libgap(ZZ)
sage: type(i)
<class 'sage.libs.gap.element.GapElement_Ring'>

Python

>>> from sage.all import *
>>> i = libgap(ZZ)
>>> type(i)
<class 'sage.libs.gap.element.GapElement_Ring'>

ring_cyclotomic()[source]¶

Construct an integer ring.

EXAMPLES:

Sage

sage: libgap.CyclotomicField(6).ring_cyclotomic()
Cyclotomic Field of order 3 and degree 2

Python

>>> from sage.all import *
>>> libgap.CyclotomicField(Integer(6)).ring_cyclotomic()
Cyclotomic Field of order 3 and degree 2

ring_finite_field(var='a')[source]¶

Construct an integer ring.

EXAMPLES:

Sage

sage: libgap.GF(3,2).ring_finite_field(var='A')
Finite Field in A of size 3^2

Python

>>> from sage.all import *
>>> libgap.GF(Integer(3),Integer(2)).ring_finite_field(var='A')
Finite Field in A of size 3^2

ring_integer()[source]¶

Construct the Sage integers.

EXAMPLES:

Sage

sage: libgap.eval('Integers').ring_integer()
Integer Ring

Python

>>> from sage.all import *
>>> libgap.eval('Integers').ring_integer()
Integer Ring

ring_integer_mod()[source]¶

Construct a Sage integer mod ring.

EXAMPLES:

Sage

sage: libgap.eval('ZmodnZ(15)').ring_integer_mod()
Ring of integers modulo 15

Python

>>> from sage.all import *
>>> libgap.eval('ZmodnZ(15)').ring_integer_mod()
Ring of integers modulo 15

ring_polynomial()[source]¶

Construct a polynomial ring.

EXAMPLES:

Sage

sage: B = libgap(QQ['x'])
sage: B.ring_polynomial()
Univariate Polynomial Ring in x over Rational Field

sage: B = libgap(ZZ['x','y'])
sage: B.ring_polynomial()
Multivariate Polynomial Ring in x, y over Integer Ring

Python

>>> from sage.all import *
>>> B = libgap(QQ['x'])
>>> B.ring_polynomial()
Univariate Polynomial Ring in x over Rational Field

>>> B = libgap(ZZ['x','y'])
>>> B.ring_polynomial()
Multivariate Polynomial Ring in x, y over Integer Ring

ring_rational()[source]¶

Construct the Sage rationals.

EXAMPLES:

Sage

sage: libgap.eval('Rationals').ring_rational()
Rational Field

Python

>>> from sage.all import *
>>> libgap.eval('Rationals').ring_rational()
Rational Field

sage(**kwds)[source]¶

Return the Sage equivalent of the GapElement_Ring.

INPUT:

**kwds – keywords that are passed on to the ring_ method

OUTPUT: a Sage ring

EXAMPLES:

Sage

sage: libgap.eval('Integers').sage()
Integer Ring

sage: libgap.eval('Rationals').sage()
Rational Field

sage: libgap.eval('ZmodnZ(15)').sage()
Ring of integers modulo 15

sage: libgap.GF(3,2).sage(var='A')
Finite Field in A of size 3^2

sage: libgap.CyclotomicField(6).sage()
Cyclotomic Field of order 3 and degree 2

sage: libgap(QQ['x','y']).sage()
Multivariate Polynomial Ring in x, y over Rational Field

Python

>>> from sage.all import *
>>> libgap.eval('Integers').sage()
Integer Ring

>>> libgap.eval('Rationals').sage()
Rational Field

>>> libgap.eval('ZmodnZ(15)').sage()
Ring of integers modulo 15

>>> libgap.GF(Integer(3),Integer(2)).sage(var='A')
Finite Field in A of size 3^2

>>> libgap.CyclotomicField(Integer(6)).sage()
Cyclotomic Field of order 3 and degree 2

>>> libgap(QQ['x','y']).sage()
Multivariate Polynomial Ring in x, y over Rational Field

class sage.libs.gap.element.GapElement_String[source]¶

Bases: GapElement

Derived class of GapElement for GAP strings.

EXAMPLES:

Sage

sage: s = libgap('string')
sage: type(s)
<class 'sage.libs.gap.element.GapElement_String'>
sage: s
"string"
sage: print(s)
string

Python

>>> from sage.all import *
>>> s = libgap('string')
>>> type(s)
<class 'sage.libs.gap.element.GapElement_String'>
>>> s
"string"
>>> print(s)
string

sage()[source]¶

Convert this GapElement_String to a Python string.

OUTPUT: a Python string

EXAMPLES:

Sage

sage: s = libgap.eval(' "string" '); s
"string"
sage: type(_)
<class 'sage.libs.gap.element.GapElement_String'>
sage: str(s)
'string'
sage: s.sage()
'string'
sage: type(_)
<class 'str'>

Python

>>> from sage.all import *
>>> s = libgap.eval(' "string" '); s
"string"
>>> type(_)
<class 'sage.libs.gap.element.GapElement_String'>
>>> str(s)
'string'
>>> s.sage()
'string'
>>> type(_)
<class 'str'>