Finitely Presented Groups#

Finitely presented groups are constructed as quotients of free_group:

sage: F.<a,b,c> = FreeGroup()
sage: G = F / [a^2, b^2, c^2, a*b*c*a*b*c]
sage: G
Finitely presented group < a, b, c | a^2, b^2, c^2, (a*b*c)^2 >

>>> from sage.all import *
>>> F = FreeGroup(names=('a', 'b', 'c',)); (a, b, c,) = F._first_ngens(3)
>>> G = F / [a**Integer(2), b**Integer(2), c**Integer(2), a*b*c*a*b*c]
>>> G
Finitely presented group < a, b, c | a^2, b^2, c^2, (a*b*c)^2 >

One can create their elements by multiplying the generators or by specifying a Tietze list (see Tietze()) as in the case of free groups:

sage: G.gen(0) * G.gen(1)
a*b
sage: G([1,2,-1])
a*b*a^-1
sage: a.parent()
Free Group on generators {a, b, c}
sage: G.inject_variables()
Defining a, b, c
sage: a.parent()
Finitely presented group < a, b, c | a^2, b^2, c^2, (a*b*c)^2 >

>>> from sage.all import *
>>> G.gen(Integer(0)) * G.gen(Integer(1))
a*b
>>> G([Integer(1),Integer(2),-Integer(1)])
a*b*a^-1
>>> a.parent()
Free Group on generators {a, b, c}
>>> G.inject_variables()
Defining a, b, c
>>> a.parent()
Finitely presented group < a, b, c | a^2, b^2, c^2, (a*b*c)^2 >

Notice that, even if they are represented in the same way, the elements of a finitely presented group and the elements of the corresponding free group are not the same thing. However, they can be converted from one parent to the other:

sage: F.<a,b,c> = FreeGroup()
sage: G = F / [a^2,b^2,c^2,a*b*c*a*b*c]
sage: F([1])
a
sage: G([1])
a
sage: F([1]) is G([1])
False
sage: F([1]) == G([1])
False
sage: G(a*b/c)
a*b*c^-1
sage: F(G(a*b/c))
a*b*c^-1

>>> from sage.all import *
>>> F = FreeGroup(names=('a', 'b', 'c',)); (a, b, c,) = F._first_ngens(3)
>>> G = F / [a**Integer(2),b**Integer(2),c**Integer(2),a*b*c*a*b*c]
>>> F([Integer(1)])
a
>>> G([Integer(1)])
a
>>> F([Integer(1)]) is G([Integer(1)])
False
>>> F([Integer(1)]) == G([Integer(1)])
False
>>> G(a*b/c)
a*b*c^-1
>>> F(G(a*b/c))
a*b*c^-1

Finitely presented groups are implemented via GAP. You can use the gap() method to access the underlying LibGAP object:

sage: G = FreeGroup(2)
sage: G.inject_variables()
Defining x0, x1
sage: H = G / (x0^2, (x0*x1)^2, x1^2)
sage: H.gap()
<fp group on the generators [ x0, x1 ]>

>>> from sage.all import *
>>> G = FreeGroup(Integer(2))
>>> G.inject_variables()
Defining x0, x1
>>> H = G / (x0**Integer(2), (x0*x1)**Integer(2), x1**Integer(2))
>>> H.gap()
<fp group on the generators [ x0, x1 ]>

This can be useful, for example, to use GAP functions that are not yet wrapped in Sage:

sage: H.gap().LowerCentralSeries()
[ Group(<fp, no generators known>), Group(<fp, no generators known>) ]

>>> from sage.all import *
>>> H.gap().LowerCentralSeries()
[ Group(<fp, no generators known>), Group(<fp, no generators known>) ]

The same holds for the group elements:

sage: G = FreeGroup(2)
sage: H = G / (G([1, 1]), G([2, 2, 2]), G([1, 2, -1, -2]));  H
Finitely presented group < x0, x1 | x0^2, x1^3, x0*x1*x0^-1*x1^-1 >
sage: a = H([1])
sage: a
x0
sage: a.gap()
x0
sage: a.gap().Order()
2
sage: type(_)    # note that the above output is not a Sage integer
<class 'sage.libs.gap.element.GapElement_Integer'>

>>> from sage.all import *
>>> G = FreeGroup(Integer(2))
>>> H = G / (G([Integer(1), Integer(1)]), G([Integer(2), Integer(2), Integer(2)]), G([Integer(1), Integer(2), -Integer(1), -Integer(2)]));  H
Finitely presented group < x0, x1 | x0^2, x1^3, x0*x1*x0^-1*x1^-1 >
>>> a = H([Integer(1)])
>>> a
x0
>>> a.gap()
x0
>>> a.gap().Order()
2
>>> type(_)    # note that the above output is not a Sage integer
<class 'sage.libs.gap.element.GapElement_Integer'>

You can use call syntax to replace the generators with a set of arbitrary ring elements. For example, take the free abelian group obtained by modding out the commutator subgroup of the free group:

sage: G = FreeGroup(2)
sage: G_ab = G / [G([1, 2, -1, -2])];  G_ab
Finitely presented group < x0, x1 | x0*x1*x0^-1*x1^-1 >
sage: a,b = G_ab.gens()
sage: g =  a * b
sage: M1 = matrix([[1,0],[0,2]])
sage: M2 = matrix([[0,1],[1,0]])
sage: g(3, 5)
15
sage: g(M1, M1)
[1 0]
[0 4]
sage: M1*M2 == M2*M1   # matrices do not commute
False
sage: g(M1, M2)
Traceback (most recent call last):
...
ValueError: the values do not satisfy all relations of the group

>>> from sage.all import *
>>> G = FreeGroup(Integer(2))
>>> G_ab = G / [G([Integer(1), Integer(2), -Integer(1), -Integer(2)])];  G_ab
Finitely presented group < x0, x1 | x0*x1*x0^-1*x1^-1 >
>>> a,b = G_ab.gens()
>>> g =  a * b
>>> M1 = matrix([[Integer(1),Integer(0)],[Integer(0),Integer(2)]])
>>> M2 = matrix([[Integer(0),Integer(1)],[Integer(1),Integer(0)]])
>>> g(Integer(3), Integer(5))
15
>>> g(M1, M1)
[1 0]
[0 4]
>>> M1*M2 == M2*M1   # matrices do not commute
False
>>> g(M1, M2)
Traceback (most recent call last):
...
ValueError: the values do not satisfy all relations of the group

Warning

Some methods are not guaranteed to finish since the word problem for finitely presented groups is, in general, undecidable. In those cases the process may run until the available memory is exhausted.

REFERENCES:

AUTHOR:

  • Miguel Angel Marco Buzunariz

class sage.groups.finitely_presented.FinitelyPresentedGroup(free_group, relations, category=None)[source]#

Bases: GroupMixinLibGAP, UniqueRepresentation, Group, ParentLibGAP

A class that wraps GAP’s Finitely Presented Groups.

Warning

You should use quotient() to construct finitely presented groups as quotients of free groups.

EXAMPLES:

sage: G.<a,b> = FreeGroup()
sage: H = G / [a, b^3]
sage: H
Finitely presented group < a, b | a, b^3 >
sage: H.gens()
(a, b)

sage: F.<a,b> = FreeGroup('a, b')
sage: J = F / (F([1]), F([2, 2, 2]))
sage: J is H
True

sage: G = FreeGroup(2)
sage: H = G / (G([1, 1]), G([2, 2, 2]))
sage: H.gens()
(x0, x1)
sage: H.gen(0)
x0
sage: H.ngens()
2
sage: H.gap()
<fp group on the generators [ x0, x1 ]>
sage: type(_)
<class 'sage.libs.gap.element.GapElement'>

>>> from sage.all import *
>>> G = FreeGroup(names=('a', 'b',)); (a, b,) = G._first_ngens(2)
>>> H = G / [a, b**Integer(3)]
>>> H
Finitely presented group < a, b | a, b^3 >
>>> H.gens()
(a, b)

>>> F = FreeGroup('a, b', names=('a', 'b',)); (a, b,) = F._first_ngens(2)
>>> J = F / (F([Integer(1)]), F([Integer(2), Integer(2), Integer(2)]))
>>> J is H
True

>>> G = FreeGroup(Integer(2))
>>> H = G / (G([Integer(1), Integer(1)]), G([Integer(2), Integer(2), Integer(2)]))
>>> H.gens()
(x0, x1)
>>> H.gen(Integer(0))
x0
>>> H.ngens()
2
>>> H.gap()
<fp group on the generators [ x0, x1 ]>
>>> type(_)
<class 'sage.libs.gap.element.GapElement'>
Element[source]#

alias of FinitelyPresentedGroupElement

abelian_alexander_matrix(ring=Rational Field, simplified=True)[source]#

Return the Alexander matrix of the group with values in the group algebra of the abelianized.

INPUT:

  • ring – (default: QQ) the base ring of the group algebra

  • simplified – boolean (default: False); if set to True use Gauss elimination and erase rows and columns

OUTPUT:

  • A – a matrix with coefficients in R

  • ideal – an list of generators of an ideal I of R = A.base_ring() such that R/I is the group algebra of the abelianization of self

EXAMPLES:

sage: G.<a,b,c> = FreeGroup()
sage: H = G.quotient([a*b/a/b, a*c/a/c, c*b/c/b])
sage: A, ideal = H.abelian_alexander_matrix()
sage: A
[-f2 + 1  f1 - 1       0]
[-f3 + 1       0  f1 - 1]
[      0  f3 - 1 -f2 + 1]
sage: A.base_ring()
Multivariate Laurent Polynomial Ring in f1, f2, f3 over Rational Field
sage: ideal
[]
sage: G = FreeGroup(3)/[(2, 1, 1), (1, 2, 2, 3, 3)]
sage: A, ideal = G.abelian_alexander_matrix(simplified=True); A
[-f3^2 - f3^4 - f3^6         f3^3 + f3^6]
sage: g = FreeGroup(1) / []
sage: g.abelian_alexander_matrix()
([], [])
sage: g.abelian_alexander_matrix()[0].base_ring()
Univariate Laurent Polynomial Ring in f1 over Rational Field
sage: g = FreeGroup(0) / []
sage: A, ideal = g.abelian_alexander_matrix(); A
[]
sage: A.base_ring()
Rational Field

>>> from sage.all import *
>>> G = FreeGroup(names=('a', 'b', 'c',)); (a, b, c,) = G._first_ngens(3)
>>> H = G.quotient([a*b/a/b, a*c/a/c, c*b/c/b])
>>> A, ideal = H.abelian_alexander_matrix()
>>> A
[-f2 + 1  f1 - 1       0]
[-f3 + 1       0  f1 - 1]
[      0  f3 - 1 -f2 + 1]
>>> A.base_ring()
Multivariate Laurent Polynomial Ring in f1, f2, f3 over Rational Field
>>> ideal
[]
>>> G = FreeGroup(Integer(3))/[(Integer(2), Integer(1), Integer(1)), (Integer(1), Integer(2), Integer(2), Integer(3), Integer(3))]
>>> A, ideal = G.abelian_alexander_matrix(simplified=True); A
[-f3^2 - f3^4 - f3^6         f3^3 + f3^6]
>>> g = FreeGroup(Integer(1)) / []
>>> g.abelian_alexander_matrix()
([], [])
>>> g.abelian_alexander_matrix()[Integer(0)].base_ring()
Univariate Laurent Polynomial Ring in f1 over Rational Field
>>> g = FreeGroup(Integer(0)) / []
>>> A, ideal = g.abelian_alexander_matrix(); A
[]
>>> A.base_ring()
Rational Field
abelian_invariants()[source]#

Return the abelian invariants of self.

The abelian invariants are given by a list of integers \((i_1, \ldots, i_j)\), such that the abelianization of the group is isomorphic to \(\ZZ / (i_1) \times \cdots \times \ZZ / (i_j)\).

EXAMPLES:

sage: G = FreeGroup(4, 'g')
sage: G.inject_variables()
Defining g0, g1, g2, g3
sage: H = G.quotient([g1^2, g2*g1*g2^(-1)*g1^(-1), g1*g3^(-2), g0^4])
sage: H.abelian_invariants()
(0, 4, 4)

>>> from sage.all import *
>>> G = FreeGroup(Integer(4), 'g')
>>> G.inject_variables()
Defining g0, g1, g2, g3
>>> H = G.quotient([g1**Integer(2), g2*g1*g2**(-Integer(1))*g1**(-Integer(1)), g1*g3**(-Integer(2)), g0**Integer(4)])
>>> H.abelian_invariants()
(0, 4, 4)

ALGORITHM:

Uses GAP.

abelianization_map()[source]#

Return the abelianization map of self.

OUTPUT: the abelianization map of self as a homomorphism of finitely presented groups

EXAMPLES:

sage: G = FreeGroup(4, 'g')
sage: G.inject_variables(verbose=False)
sage: H = G.quotient([g1^2, g2*g1*g2^(-1)*g1^(-1), g1*g3^(-2), g0^4])
sage: H.abelianization_map()
Group morphism:
    From: Finitely presented group  < g0, g1, g2, g3 | g1^2, g2*g1*g2^-1*g1^-1, g1*g3^-2, g0^4 >
    To:   Finitely presented group  < f2, f3, f4 | f2^-1*f3^-1*f2*f3, f2^-1*f4^-1*f2*f4, f3^-1*f4^-1*f3*f4, f2^4, f3^4 >
sage: g = FreeGroup(0) / []
sage: g.abelianization_map()
Group endomorphism of Finitely presented group  <  |  >

>>> from sage.all import *
>>> G = FreeGroup(Integer(4), 'g')
>>> G.inject_variables(verbose=False)
>>> H = G.quotient([g1**Integer(2), g2*g1*g2**(-Integer(1))*g1**(-Integer(1)), g1*g3**(-Integer(2)), g0**Integer(4)])
>>> H.abelianization_map()
Group morphism:
    From: Finitely presented group  < g0, g1, g2, g3 | g1^2, g2*g1*g2^-1*g1^-1, g1*g3^-2, g0^4 >
    To:   Finitely presented group  < f2, f3, f4 | f2^-1*f3^-1*f2*f3, f2^-1*f4^-1*f2*f4, f3^-1*f4^-1*f3*f4, f2^4, f3^4 >
>>> g = FreeGroup(Integer(0)) / []
>>> g.abelianization_map()
Group endomorphism of Finitely presented group  <  |  >
abelianization_to_algebra(ring=Rational Field)[source]#

Return the group algebra of the abelianization of self together with the monomials representing the generators of self.

INPUT:

  • ring – (default: QQ); the base ring for the group algebra of self

OUTPUT:

  • ab – the abelianization of self as a finitely presented group with a minimal number \(n\) of generators.

  • R – a Laurent polynomial ring with \(n\) variables with base ring ring.

  • ideal – a list of generators of an ideal I in R such that R/I is the group algebra of the abelianization over ring

  • image – a list with the images of the generators of self in R/I

EXAMPLES:

sage: G = FreeGroup(4, 'g')
sage: G.inject_variables()
Defining g0, g1, g2, g3
sage: H = G.quotient([g1^2, g2*g1*g2^(-1)*g1^(-1), g1*g3^(-2), g0^4])
sage: H.abelianization_to_algebra()
(Finitely presented group  < f2, f3, f4 | f2^-1*f3^-1*f2*f3, f2^-1*f4^-1*f2*f4,
                                          f3^-1*f4^-1*f3*f4, f2^4, f3^4 >,
 Multivariate Laurent Polynomial Ring in f2, f3, f4 over Rational Field,
 [f2^4 - 1, f3^4 - 1], [f2^-1*f3^-2, f3^-2, f4, f3])
sage: g=FreeGroup(0) / []
sage: g.abelianization_to_algebra()
(Finitely presented group  <  |  >, Rational Field, [], [])

>>> from sage.all import *
>>> G = FreeGroup(Integer(4), 'g')
>>> G.inject_variables()
Defining g0, g1, g2, g3
>>> H = G.quotient([g1**Integer(2), g2*g1*g2**(-Integer(1))*g1**(-Integer(1)), g1*g3**(-Integer(2)), g0**Integer(4)])
>>> H.abelianization_to_algebra()
(Finitely presented group  < f2, f3, f4 | f2^-1*f3^-1*f2*f3, f2^-1*f4^-1*f2*f4,
                                          f3^-1*f4^-1*f3*f4, f2^4, f3^4 >,
 Multivariate Laurent Polynomial Ring in f2, f3, f4 over Rational Field,
 [f2^4 - 1, f3^4 - 1], [f2^-1*f3^-2, f3^-2, f4, f3])
>>> g=FreeGroup(Integer(0)) / []
>>> g.abelianization_to_algebra()
(Finitely presented group  <  |  >, Rational Field, [], [])
alexander_matrix(im_gens=None)[source]#

Return the Alexander matrix of the group.

This matrix is given by the fox derivatives of the relations with respect to the generators.

  • im_gens – (optional) the images of the generators

OUTPUT:

A matrix with coefficients in the group algebra. If im_gens is given, the coefficients will live in the same algebra as the given values. The result depends on the (fixed) choice of presentation.

EXAMPLES:

sage: G.<a,b,c> = FreeGroup()
sage: H = G.quotient([a*b/a/b, a*c/a/c, c*b/c/b])
sage: H.alexander_matrix()
[     1 - a*b*a^-1 a - a*b*a^-1*b^-1                 0]
[     1 - a*c*a^-1                 0 a - a*c*a^-1*c^-1]
[                0 c - c*b*c^-1*b^-1      1 - c*b*c^-1]

>>> from sage.all import *
>>> G = FreeGroup(names=('a', 'b', 'c',)); (a, b, c,) = G._first_ngens(3)
>>> H = G.quotient([a*b/a/b, a*c/a/c, c*b/c/b])
>>> H.alexander_matrix()
[     1 - a*b*a^-1 a - a*b*a^-1*b^-1                 0]
[     1 - a*c*a^-1                 0 a - a*c*a^-1*c^-1]
[                0 c - c*b*c^-1*b^-1      1 - c*b*c^-1]

If we introduce the images of the generators, we obtain the result in the corresponding algebra.

sage: G.<a,b,c,d,e> = FreeGroup()
sage: H = G.quotient([a*b/a/b, a*c/a/c, a*d/a/d, b*c*d/(c*d*b), b*c*d/(d*b*c)])
sage: H.alexander_matrix()
[              1 - a*b*a^-1          a - a*b*a^-1*b^-1                          0                          0                          0]
[              1 - a*c*a^-1                          0          a - a*c*a^-1*c^-1                          0                          0]
[              1 - a*d*a^-1                          0                          0          a - a*d*a^-1*d^-1                          0]
[                         0             1 - b*c*d*b^-1   b - b*c*d*b^-1*d^-1*c^-1      b*c - b*c*d*b^-1*d^-1                          0]
[                         0        1 - b*c*d*c^-1*b^-1             b - b*c*d*c^-1 b*c - b*c*d*c^-1*b^-1*d^-1                          0]
sage: R.<t1,t2,t3,t4> = LaurentPolynomialRing(ZZ)
sage: H.alexander_matrix([t1,t2,t3,t4])
[    -t2 + 1      t1 - 1           0           0           0]
[    -t3 + 1           0      t1 - 1           0           0]
[    -t4 + 1           0           0      t1 - 1           0]
[          0  -t3*t4 + 1      t2 - 1  t2*t3 - t3           0]
[          0     -t4 + 1 -t2*t4 + t2   t2*t3 - 1           0]

>>> from sage.all import *
>>> G = FreeGroup(names=('a', 'b', 'c', 'd', 'e',)); (a, b, c, d, e,) = G._first_ngens(5)
>>> H = G.quotient([a*b/a/b, a*c/a/c, a*d/a/d, b*c*d/(c*d*b), b*c*d/(d*b*c)])
>>> H.alexander_matrix()
[              1 - a*b*a^-1          a - a*b*a^-1*b^-1                          0                          0                          0]
[              1 - a*c*a^-1                          0          a - a*c*a^-1*c^-1                          0                          0]
[              1 - a*d*a^-1                          0                          0          a - a*d*a^-1*d^-1                          0]
[                         0             1 - b*c*d*b^-1   b - b*c*d*b^-1*d^-1*c^-1      b*c - b*c*d*b^-1*d^-1                          0]
[                         0        1 - b*c*d*c^-1*b^-1             b - b*c*d*c^-1 b*c - b*c*d*c^-1*b^-1*d^-1                          0]
>>> R = LaurentPolynomialRing(ZZ, names=('t1', 't2', 't3', 't4',)); (t1, t2, t3, t4,) = R._first_ngens(4)
>>> H.alexander_matrix([t1,t2,t3,t4])
[    -t2 + 1      t1 - 1           0           0           0]
[    -t3 + 1           0      t1 - 1           0           0]
[    -t4 + 1           0           0      t1 - 1           0]
[          0  -t3*t4 + 1      t2 - 1  t2*t3 - t3           0]
[          0     -t4 + 1 -t2*t4 + t2   t2*t3 - 1           0]
as_permutation_group(limit=4096000)[source]#

Return an isomorphic permutation group.

The generators of the resulting group correspond to the images by the isomorphism of the generators of the given group.

INPUT:

  • limit – integer (default: 4096000); the maximal number of cosets before the computation is aborted

OUTPUT:

A Sage PermutationGroup(). If the number of cosets exceeds the given limit, a ValueError is returned.

EXAMPLES:

sage: G.<a,b> = FreeGroup()
sage: H = G / (a^2, b^3, a*b*~a*~b)
sage: H.as_permutation_group()
Permutation Group with generators [(1,2)(3,5)(4,6), (1,3,4)(2,5,6)]

sage: G.<a,b> = FreeGroup()
sage: H = G / [a^3*b]
sage: H.as_permutation_group(limit=1000)
Traceback (most recent call last):
...
ValueError: Coset enumeration exceeded limit, is the group finite?

>>> from sage.all import *
>>> G = FreeGroup(names=('a', 'b',)); (a, b,) = G._first_ngens(2)
>>> H = G / (a**Integer(2), b**Integer(3), a*b*~a*~b)
>>> H.as_permutation_group()
Permutation Group with generators [(1,2)(3,5)(4,6), (1,3,4)(2,5,6)]

>>> G = FreeGroup(names=('a', 'b',)); (a, b,) = G._first_ngens(2)
>>> H = G / [a**Integer(3)*b]
>>> H.as_permutation_group(limit=Integer(1000))
Traceback (most recent call last):
...
ValueError: Coset enumeration exceeded limit, is the group finite?

ALGORITHM:

Uses GAP’s coset enumeration on the trivial subgroup.

Warning

This is in general not a decidable problem (in fact, it is not even possible to check if the group is finite or not). If the group is infinite, or too big, you should be prepared for a long computation that consumes all the memory without finishing if you do not set a sensible limit.

cardinality(limit=4096000)[source]#

Compute the cardinality of self.

INPUT:

  • limit – integer (default: 4096000); the maximal number of cosets before the computation is aborted

OUTPUT: integer or Infinity; the number of elements in the group

EXAMPLES:

sage: G.<a,b> = FreeGroup('a, b')
sage: H = G / (a^2, b^3, a*b*~a*~b)
sage: H.cardinality()
6

sage: F.<a,b,c> = FreeGroup()
sage: J = F / (F([1]), F([2, 2, 2]))
sage: J.cardinality()
+Infinity

>>> from sage.all import *
>>> G = FreeGroup('a, b', names=('a', 'b',)); (a, b,) = G._first_ngens(2)
>>> H = G / (a**Integer(2), b**Integer(3), a*b*~a*~b)
>>> H.cardinality()
6

>>> F = FreeGroup(names=('a', 'b', 'c',)); (a, b, c,) = F._first_ngens(3)
>>> J = F / (F([Integer(1)]), F([Integer(2), Integer(2), Integer(2)]))
>>> J.cardinality()
+Infinity

ALGORITHM:

Uses GAP.

Warning

This is in general not a decidable problem, so it is not guaranteed to give an answer. If the group is infinite, or too big, you should be prepared for a long computation that consumes all the memory without finishing if you do not set a sensible limit.

characteristic_varieties(ring=Rational Field, matrix_ideal=None, groebner=False)[source]#

Return the characteristic varieties of the group self.

There are several definitions of the characteristic varieties of a group \(G\), see e.g. [CS1999a]. Let \(\Lambda\) be the group algebra of \(G/G'\) and \(\mathbb{T}\) its associated algebraic variety (a torus). Each element \(\xi\in\mathbb{T}\) defines a local system of coefficients and the \(k\)-th characteristic variety is

\[V_k(G) = \{\xi\in\mathbb{T}\mid \dim H^1(G;\xi)\geq k\}.\]

These varieties are defined by ideals in \(\Lambda\).

INPUT:

  • ring – (default: QQ) the base ring of the group algebra

  • groebner – boolean (default: False); If set to True the minimal associated primes of the ideals and their groebner bases are computed; ignored if the base ring is not a field

OUTPUT:

A dictionary with keys the indices of the varieties. If groebner is False the values are the ideals defining the characteristic varieties. If it is True, lists for Gröbner bases for the ideal of each irreducible component, stopping when the first time a characteristic variety is empty.

EXAMPLES:

sage: L = [2*(i, j) + 2* (-i, -j) for i, j in ((1, 2), (2, 3), (3, 1))]
sage: G = FreeGroup(3) / L
sage: G.characteristic_varieties(groebner=True)
{0: [(0,)],
 1: [(f1 - 1, f2 - 1, f3 - 1), (f1*f3 + 1, f2 - 1), (f1*f2 + 1, f3 - 1), (f2*f3 + 1, f1 - 1),
     (f2*f3 + 1, f1 - f2), (f2*f3 + 1, f1 - f3), (f1*f3 + 1, f2 - f3)],
 2: [(f1 - 1, f2 - 1, f3 - 1), (f1 + 1, f2 - 1, f3 - 1), (f1 - 1, f2 - 1, f3 + 1),
     (f3^2 + 1, f1 - f3, f2 - f3), (f1 - 1, f2 + 1, f3 - 1)],
 3: [(f1 - 1, f2 - 1, f3 - 1)],
 4: []}
sage: G = FreeGroup(2)/[2*(1,2,-1,-2)]
sage: G.characteristic_varieties()
{0: Ideal (0) of Multivariate Laurent Polynomial Ring in f1, f2 over Rational Field,
 1: Ideal (f2 - 1, f1 - 1) of Multivariate Laurent Polynomial Ring in f1, f2 over Rational Field,
 2: Ideal (f2 - 1, f1 - 1) of Multivariate Laurent Polynomial Ring in f1, f2 over Rational Field,
 3: Ideal (1) of Multivariate Laurent Polynomial Ring in f1, f2 over Rational Field}
sage: G.characteristic_varieties(ring=ZZ)
{0: Ideal (0) of Multivariate Laurent Polynomial Ring in f1, f2 over Integer Ring,
 1: Ideal (2*f2 - 2, 2*f1 - 2) of Multivariate Laurent Polynomial Ring in f1, f2 over Integer Ring,
 2: Ideal (f2 - 1, f1 - 1) of Multivariate Laurent Polynomial Ring in f1, f2 over Integer Ring,
 3: Ideal (1) of Multivariate Laurent Polynomial Ring in f1, f2 over Integer Ring}
sage: G = FreeGroup(2)/[(1,2,1,-2,-1,-2)]
sage: G.characteristic_varieties()
{0: Ideal (0) of Univariate Laurent Polynomial Ring in f2 over Rational Field,
 1: Ideal (-1 + 2*f2 - 2*f2^2 + f2^3) of Univariate Laurent Polynomial Ring in f2 over Rational Field,
 2: Ideal (1) of Univariate Laurent Polynomial Ring in f2 over Rational Field}
sage: G.characteristic_varieties(groebner=True)
{0: [0], 1: [-1 + f2, 1 - f2 + f2^2], 2: []}
sage: G = FreeGroup(2)/[3 * (1, ), 2 * (2, )]
sage: G.characteristic_varieties(groebner=True)
{0: [-1 + F1, 1 + F1, 1 - F1 + F1^2, 1 + F1 + F1^2], 1: [1 - F1 + F1^2],  2: []}
sage: G = FreeGroup(2)/[2 * (2, )]
sage: G.characteristic_varieties(groebner=True)
{0: [(f1 + 1,), (f1 - 1,)], 1: [(f1 + 1,), (f1 - 1, f2 - 1)], 2: []}
sage: G = (FreeGroup(0) / [])
sage: G.characteristic_varieties()
{0: Principal ideal (0) of Rational Field,
 1: Principal ideal (1) of Rational Field}
sage: G.characteristic_varieties(groebner=True)
{0: [(0,)], 1: [(1,)]}

>>> from sage.all import *
>>> L = [Integer(2)*(i, j) + Integer(2)* (-i, -j) for i, j in ((Integer(1), Integer(2)), (Integer(2), Integer(3)), (Integer(3), Integer(1)))]
>>> G = FreeGroup(Integer(3)) / L
>>> G.characteristic_varieties(groebner=True)
{0: [(0,)],
 1: [(f1 - 1, f2 - 1, f3 - 1), (f1*f3 + 1, f2 - 1), (f1*f2 + 1, f3 - 1), (f2*f3 + 1, f1 - 1),
     (f2*f3 + 1, f1 - f2), (f2*f3 + 1, f1 - f3), (f1*f3 + 1, f2 - f3)],
 2: [(f1 - 1, f2 - 1, f3 - 1), (f1 + 1, f2 - 1, f3 - 1), (f1 - 1, f2 - 1, f3 + 1),
     (f3^2 + 1, f1 - f3, f2 - f3), (f1 - 1, f2 + 1, f3 - 1)],
 3: [(f1 - 1, f2 - 1, f3 - 1)],
 4: []}
>>> G = FreeGroup(Integer(2))/[Integer(2)*(Integer(1),Integer(2),-Integer(1),-Integer(2))]
>>> G.characteristic_varieties()
{0: Ideal (0) of Multivariate Laurent Polynomial Ring in f1, f2 over Rational Field,
 1: Ideal (f2 - 1, f1 - 1) of Multivariate Laurent Polynomial Ring in f1, f2 over Rational Field,
 2: Ideal (f2 - 1, f1 - 1) of Multivariate Laurent Polynomial Ring in f1, f2 over Rational Field,
 3: Ideal (1) of Multivariate Laurent Polynomial Ring in f1, f2 over Rational Field}
>>> G.characteristic_varieties(ring=ZZ)
{0: Ideal (0) of Multivariate Laurent Polynomial Ring in f1, f2 over Integer Ring,
 1: Ideal (2*f2 - 2, 2*f1 - 2) of Multivariate Laurent Polynomial Ring in f1, f2 over Integer Ring,
 2: Ideal (f2 - 1, f1 - 1) of Multivariate Laurent Polynomial Ring in f1, f2 over Integer Ring,
 3: Ideal (1) of Multivariate Laurent Polynomial Ring in f1, f2 over Integer Ring}
>>> G = FreeGroup(Integer(2))/[(Integer(1),Integer(2),Integer(1),-Integer(2),-Integer(1),-Integer(2))]
>>> G.characteristic_varieties()
{0: Ideal (0) of Univariate Laurent Polynomial Ring in f2 over Rational Field,
 1: Ideal (-1 + 2*f2 - 2*f2^2 + f2^3) of Univariate Laurent Polynomial Ring in f2 over Rational Field,
 2: Ideal (1) of Univariate Laurent Polynomial Ring in f2 over Rational Field}
>>> G.characteristic_varieties(groebner=True)
{0: [0], 1: [-1 + f2, 1 - f2 + f2^2], 2: []}
>>> G = FreeGroup(Integer(2))/[Integer(3) * (Integer(1), ), Integer(2) * (Integer(2), )]
>>> G.characteristic_varieties(groebner=True)
{0: [-1 + F1, 1 + F1, 1 - F1 + F1^2, 1 + F1 + F1^2], 1: [1 - F1 + F1^2],  2: []}
>>> G = FreeGroup(Integer(2))/[Integer(2) * (Integer(2), )]
>>> G.characteristic_varieties(groebner=True)
{0: [(f1 + 1,), (f1 - 1,)], 1: [(f1 + 1,), (f1 - 1, f2 - 1)], 2: []}
>>> G = (FreeGroup(Integer(0)) / [])
>>> G.characteristic_varieties()
{0: Principal ideal (0) of Rational Field,
 1: Principal ideal (1) of Rational Field}
>>> G.characteristic_varieties(groebner=True)
{0: [(0,)], 1: [(1,)]}
direct_product(H, reduced=False, new_names=True)[source]#

Return the direct product of self with finitely presented group H.

Calls GAP function DirectProduct, which returns the direct product of a list of groups of any representation.

From [Joh1990] (p. 45, proposition 4): If \(G\), \(H\) are groups presented by \(\langle X \mid R \rangle\) and \(\langle Y \mid S \rangle\) respectively, then their direct product has the presentation \(\langle X, Y \mid R, S, [X, Y] \rangle\) where \([X, Y]\) denotes the set of commutators \(\{ x^{-1} y^{-1} x y \mid x \in X, y \in Y \}\).

INPUT:

  • H – a finitely presented group

  • reduced – (default: False) boolean; if True, then attempt to reduce the presentation of the product group

  • new_names – (default: True) boolean; If True, then lexicographical variable names are assigned to the generators of the group to be returned. If False, the group to be returned keeps the generator names of the two groups forming the direct product. Note that one cannot ask to reduce the output and ask to keep the old variable names, as they may change meaning in the output group if its presentation is reduced.

OUTPUT: the direct product of self with H as a finitely presented group

EXAMPLES:

sage: G = FreeGroup()
sage: C12 =  ( G / [G([1,1,1,1])] ).direct_product( G / [G([1,1,1])]); C12
Finitely presented group < a, b | a^4, b^3, a^-1*b^-1*a*b >
sage: C12.order(), C12.as_permutation_group().is_cyclic()
(12, True)
sage: klein = ( G / [G([1,1])] ).direct_product( G / [G([1,1])]); klein
Finitely presented group < a, b | a^2, b^2, a^-1*b^-1*a*b >
sage: klein.order(), klein.as_permutation_group().is_cyclic()
(4, False)

>>> from sage.all import *
>>> G = FreeGroup()
>>> C12 =  ( G / [G([Integer(1),Integer(1),Integer(1),Integer(1)])] ).direct_product( G / [G([Integer(1),Integer(1),Integer(1)])]); C12
Finitely presented group < a, b | a^4, b^3, a^-1*b^-1*a*b >
>>> C12.order(), C12.as_permutation_group().is_cyclic()
(12, True)
>>> klein = ( G / [G([Integer(1),Integer(1)])] ).direct_product( G / [G([Integer(1),Integer(1)])]); klein
Finitely presented group < a, b | a^2, b^2, a^-1*b^-1*a*b >
>>> klein.order(), klein.as_permutation_group().is_cyclic()
(4, False)

We can keep the variable names from self and H to examine how new relations are formed:

sage: F = FreeGroup("a"); G = FreeGroup("g")
sage: X = G / [G.0^12]; A = F / [F.0^6]
sage: X.direct_product(A, new_names=False)
Finitely presented group < g, a | g^12, a^6, g^-1*a^-1*g*a >
sage: A.direct_product(X, new_names=False)
Finitely presented group < a, g | a^6, g^12, a^-1*g^-1*a*g >

>>> from sage.all import *
>>> F = FreeGroup("a"); G = FreeGroup("g")
>>> X = G / [G.gen(0)**Integer(12)]; A = F / [F.gen(0)**Integer(6)]
>>> X.direct_product(A, new_names=False)
Finitely presented group < g, a | g^12, a^6, g^-1*a^-1*g*a >
>>> A.direct_product(X, new_names=False)
Finitely presented group < a, g | a^6, g^12, a^-1*g^-1*a*g >

Or we can attempt to reduce the output group presentation:

sage: F = FreeGroup("a"); G = FreeGroup("g")
sage: X = G / [G.0]; A = F / [F.0]
sage: X.direct_product(A, new_names=True)
Finitely presented group < a, b | a, b, a^-1*b^-1*a*b >
sage: X.direct_product(A, reduced=True, new_names=True)
Finitely presented group <  |  >

>>> from sage.all import *
>>> F = FreeGroup("a"); G = FreeGroup("g")
>>> X = G / [G.gen(0)]; A = F / [F.gen(0)]
>>> X.direct_product(A, new_names=True)
Finitely presented group < a, b | a, b, a^-1*b^-1*a*b >
>>> X.direct_product(A, reduced=True, new_names=True)
Finitely presented group <  |  >

But we cannot do both:

sage: K = FreeGroup(['a','b'])
sage: D = K / [K.0^5, K.1^8]
sage: D.direct_product(D, reduced=True, new_names=False)
Traceback (most recent call last):
...
ValueError: cannot reduce output and keep old variable names

>>> from sage.all import *
>>> K = FreeGroup(['a','b'])
>>> D = K / [K.gen(0)**Integer(5), K.gen(1)**Integer(8)]
>>> D.direct_product(D, reduced=True, new_names=False)
Traceback (most recent call last):
...
ValueError: cannot reduce output and keep old variable names

AUTHORS:

  • Davis Shurbert (2013-07-20): initial version

epimorphisms(H)[source]#

Return the epimorphisms from self to \(H\), up to automorphism of \(H\).

INPUT:

  • H – another group

EXAMPLES:

sage: F = FreeGroup(3)
sage: G = F / [F([1, 2, 3, 1, 2, 3]), F([1, 1, 1])]
sage: H = AlternatingGroup(3)
sage: G.epimorphisms(H)
[Generic morphism:
   From: Finitely presented group < x0, x1, x2 | x0*x1*x2*x0*x1*x2, x0^3 >
   To:   Alternating group of order 3!/2 as a permutation group
   Defn: x0 |--> ()
         x1 |--> (1,3,2)
         x2 |--> (1,2,3),
 Generic morphism:
   From: Finitely presented group < x0, x1, x2 | x0*x1*x2*x0*x1*x2, x0^3 >
   To:   Alternating group of order 3!/2 as a permutation group
   Defn: x0 |--> (1,3,2)
         x1 |--> ()
         x2 |--> (1,2,3),
 Generic morphism:
   From: Finitely presented group < x0, x1, x2 | x0*x1*x2*x0*x1*x2, x0^3 >
   To:   Alternating group of order 3!/2 as a permutation group
   Defn: x0 |--> (1,3,2)
         x1 |--> (1,2,3)
         x2 |--> (),
 Generic morphism:
   From: Finitely presented group < x0, x1, x2 | x0*x1*x2*x0*x1*x2, x0^3 >
   To:   Alternating group of order 3!/2 as a permutation group
   Defn: x0 |--> (1,2,3)
         x1 |--> (1,2,3)
         x2 |--> (1,2,3)]

>>> from sage.all import *
>>> F = FreeGroup(Integer(3))
>>> G = F / [F([Integer(1), Integer(2), Integer(3), Integer(1), Integer(2), Integer(3)]), F([Integer(1), Integer(1), Integer(1)])]
>>> H = AlternatingGroup(Integer(3))
>>> G.epimorphisms(H)
[Generic morphism:
   From: Finitely presented group < x0, x1, x2 | x0*x1*x2*x0*x1*x2, x0^3 >
   To:   Alternating group of order 3!/2 as a permutation group
   Defn: x0 |--> ()
         x1 |--> (1,3,2)
         x2 |--> (1,2,3),
 Generic morphism:
   From: Finitely presented group < x0, x1, x2 | x0*x1*x2*x0*x1*x2, x0^3 >
   To:   Alternating group of order 3!/2 as a permutation group
   Defn: x0 |--> (1,3,2)
         x1 |--> ()
         x2 |--> (1,2,3),
 Generic morphism:
   From: Finitely presented group < x0, x1, x2 | x0*x1*x2*x0*x1*x2, x0^3 >
   To:   Alternating group of order 3!/2 as a permutation group
   Defn: x0 |--> (1,3,2)
         x1 |--> (1,2,3)
         x2 |--> (),
 Generic morphism:
   From: Finitely presented group < x0, x1, x2 | x0*x1*x2*x0*x1*x2, x0^3 >
   To:   Alternating group of order 3!/2 as a permutation group
   Defn: x0 |--> (1,2,3)
         x1 |--> (1,2,3)
         x2 |--> (1,2,3)]

ALGORITHM:

Uses libgap’s GQuotients function.

free_group()[source]#

Return the free group (without relations).

OUTPUT: a FreeGroup()

EXAMPLES:

sage: G.<a,b,c> = FreeGroup()
sage: H = G / (a^2, b^3, a*b*~a*~b)
sage: H.free_group()
Free Group on generators {a, b, c}
sage: H.free_group() is G
True

>>> from sage.all import *
>>> G = FreeGroup(names=('a', 'b', 'c',)); (a, b, c,) = G._first_ngens(3)
>>> H = G / (a**Integer(2), b**Integer(3), a*b*~a*~b)
>>> H.free_group()
Free Group on generators {a, b, c}
>>> H.free_group() is G
True
order(limit=4096000)[source]#

Compute the cardinality of self.

INPUT:

  • limit – integer (default: 4096000); the maximal number of cosets before the computation is aborted

OUTPUT: integer or Infinity; the number of elements in the group

EXAMPLES:

sage: G.<a,b> = FreeGroup('a, b')
sage: H = G / (a^2, b^3, a*b*~a*~b)
sage: H.cardinality()
6

sage: F.<a,b,c> = FreeGroup()
sage: J = F / (F([1]), F([2, 2, 2]))
sage: J.cardinality()
+Infinity

>>> from sage.all import *
>>> G = FreeGroup('a, b', names=('a', 'b',)); (a, b,) = G._first_ngens(2)
>>> H = G / (a**Integer(2), b**Integer(3), a*b*~a*~b)
>>> H.cardinality()
6

>>> F = FreeGroup(names=('a', 'b', 'c',)); (a, b, c,) = F._first_ngens(3)
>>> J = F / (F([Integer(1)]), F([Integer(2), Integer(2), Integer(2)]))
>>> J.cardinality()
+Infinity

ALGORITHM:

Uses GAP.

Warning

This is in general not a decidable problem, so it is not guaranteed to give an answer. If the group is infinite, or too big, you should be prepared for a long computation that consumes all the memory without finishing if you do not set a sensible limit.

relations()[source]#

Return the relations of the group.

OUTPUT: the relations as a tuple of elements of free_group()

EXAMPLES:

sage: F = FreeGroup(5, 'x')
sage: F.inject_variables()
Defining x0, x1, x2, x3, x4
sage: G = F.quotient([x0*x2, x3*x1*x3, x2*x1*x2])
sage: G.relations()
(x0*x2, x3*x1*x3, x2*x1*x2)
sage: all(rel in F for rel in G.relations())
True

>>> from sage.all import *
>>> F = FreeGroup(Integer(5), 'x')
>>> F.inject_variables()
Defining x0, x1, x2, x3, x4
>>> G = F.quotient([x0*x2, x3*x1*x3, x2*x1*x2])
>>> G.relations()
(x0*x2, x3*x1*x3, x2*x1*x2)
>>> all(rel in F for rel in G.relations())
True
rewriting_system()[source]#

Return the rewriting system corresponding to the finitely presented group. This rewriting system can be used to reduce words with respect to the relations.

If the rewriting system is transformed into a confluent one, the reduction process will give as a result the (unique) reduced form of an element.

EXAMPLES:

sage: F.<a,b> = FreeGroup()
sage: G = F / [a^2,b^3,(a*b/a)^3,b*a*b*a]
sage: k = G.rewriting_system()
sage: k
Rewriting system of Finitely presented group < a, b | a^2, b^3, a*b^3*a^-1, b*a*b*a >
with rules:
    a^2    --->    1
    b^3    --->    1
    b*a*b*a    --->    1
    a*b^3*a^-1    --->    1

sage: G([1,1,2,2,2])
a^2*b^3
sage: k.reduce(G([1,1,2,2,2]))
1
sage: k.reduce(G([2,2,1]))
b^2*a
sage: k.make_confluent()
sage: k.reduce(G([2,2,1]))
a*b

>>> from sage.all import *
>>> F = FreeGroup(names=('a', 'b',)); (a, b,) = F._first_ngens(2)
>>> G = F / [a**Integer(2),b**Integer(3),(a*b/a)**Integer(3),b*a*b*a]
>>> k = G.rewriting_system()
>>> k
Rewriting system of Finitely presented group < a, b | a^2, b^3, a*b^3*a^-1, b*a*b*a >
with rules:
    a^2    --->    1
    b^3    --->    1
    b*a*b*a    --->    1
    a*b^3*a^-1    --->    1

>>> G([Integer(1),Integer(1),Integer(2),Integer(2),Integer(2)])
a^2*b^3
>>> k.reduce(G([Integer(1),Integer(1),Integer(2),Integer(2),Integer(2)]))
1
>>> k.reduce(G([Integer(2),Integer(2),Integer(1)]))
b^2*a
>>> k.make_confluent()
>>> k.reduce(G([Integer(2),Integer(2),Integer(1)]))
a*b
semidirect_product(H, hom, check=True, reduced=False)[source]#

The semidirect product of self with H via hom.

If there exists a homomorphism \(\phi\) from a group \(G\) to the automorphism group of a group \(H\), then we can define the semidirect product of \(G\) with \(H\) via \(\phi\) as the Cartesian product of \(G\) and \(H\) with the operation

\[(g_1, h_1)(g_2, h_2) = (g_1 g_2, \phi(g_2)(h_1) h_2).\]

INPUT:

  • H – Finitely presented group which is implicitly acted on by self and can be naturally embedded as a normal subgroup of the semidirect product.

  • hom – Homomorphism from self to the automorphism group of H. Given as a pair, with generators of self in the first slot and the images of the corresponding generators in the second. These images must be automorphisms of H, given again as a pair of generators and images.

  • check – Boolean (default True). If False the defining homomorphism and automorphism images are not tested for validity. This test can be costly with large groups, so it can be bypassed if the user is confident that his morphisms are valid.

  • reduced – Boolean (default False). If True then the method attempts to reduce the presentation of the output group.

OUTPUT:

The semidirect product of self with H via hom as a finitely presented group. See PermutationGroup_generic.semidirect_product for a more in depth explanation of a semidirect product.

AUTHORS:

  • Davis Shurbert (8-1-2013)

EXAMPLES:

Group of order 12 as two isomorphic semidirect products:

sage: D4 = groups.presentation.Dihedral(4)
sage: C3 = groups.presentation.Cyclic(3)
sage: alpha1 = ([C3.gen(0)],[C3.gen(0)])
sage: alpha2 = ([C3.gen(0)],[C3([1,1])])
sage: S1 = D4.semidirect_product(C3, ([D4.gen(1), D4.gen(0)],[alpha1,alpha2]))
sage: C2 = groups.presentation.Cyclic(2)
sage: Q = groups.presentation.DiCyclic(3)
sage: a = Q([1]); b = Q([-2])
sage: alpha = (Q.gens(), [a,b])
sage: S2 = C2.semidirect_product(Q, ([C2.0],[alpha]))
sage: S1.is_isomorphic(S2)
#I  Forcing finiteness test
True

>>> from sage.all import *
>>> D4 = groups.presentation.Dihedral(Integer(4))
>>> C3 = groups.presentation.Cyclic(Integer(3))
>>> alpha1 = ([C3.gen(Integer(0))],[C3.gen(Integer(0))])
>>> alpha2 = ([C3.gen(Integer(0))],[C3([Integer(1),Integer(1)])])
>>> S1 = D4.semidirect_product(C3, ([D4.gen(Integer(1)), D4.gen(Integer(0))],[alpha1,alpha2]))
>>> C2 = groups.presentation.Cyclic(Integer(2))
>>> Q = groups.presentation.DiCyclic(Integer(3))
>>> a = Q([Integer(1)]); b = Q([-Integer(2)])
>>> alpha = (Q.gens(), [a,b])
>>> S2 = C2.semidirect_product(Q, ([C2.gen(0)],[alpha]))
>>> S1.is_isomorphic(S2)
#I  Forcing finiteness test
True

Dihedral groups can be constructed as semidirect products of cyclic groups:

sage: C2 = groups.presentation.Cyclic(2)
sage: C8 = groups.presentation.Cyclic(8)
sage: hom = (C2.gens(), [ ([C8([1])], [C8([-1])]) ])
sage: D = C2.semidirect_product(C8, hom)
sage: D.as_permutation_group().is_isomorphic(DihedralGroup(8))
True

>>> from sage.all import *
>>> C2 = groups.presentation.Cyclic(Integer(2))
>>> C8 = groups.presentation.Cyclic(Integer(8))
>>> hom = (C2.gens(), [ ([C8([Integer(1)])], [C8([-Integer(1)])]) ])
>>> D = C2.semidirect_product(C8, hom)
>>> D.as_permutation_group().is_isomorphic(DihedralGroup(Integer(8)))
True

You can attempt to reduce the presentation of the output group:

sage: D = C2.semidirect_product(C8, hom); D
Finitely presented group < a, b | a^2, b^8, a^-1*b*a*b >
sage: D = C2.semidirect_product(C8, hom, reduced=True); D
Finitely presented group < a, b | a^2, a*b*a*b, b^8 >

sage: C3 = groups.presentation.Cyclic(3)
sage: C4 = groups.presentation.Cyclic(4)
sage: hom = (C3.gens(), [(C4.gens(), C4.gens())])
sage: C3.semidirect_product(C4, hom)
Finitely presented group < a, b | a^3, b^4, a^-1*b*a*b^-1 >
sage: D = C3.semidirect_product(C4, hom, reduced=True); D
Finitely presented group < a, b | a^3, b^4, a^-1*b*a*b^-1 >
sage: D.as_permutation_group().is_cyclic()
True

>>> from sage.all import *
>>> D = C2.semidirect_product(C8, hom); D
Finitely presented group < a, b | a^2, b^8, a^-1*b*a*b >
>>> D = C2.semidirect_product(C8, hom, reduced=True); D
Finitely presented group < a, b | a^2, a*b*a*b, b^8 >

>>> C3 = groups.presentation.Cyclic(Integer(3))
>>> C4 = groups.presentation.Cyclic(Integer(4))
>>> hom = (C3.gens(), [(C4.gens(), C4.gens())])
>>> C3.semidirect_product(C4, hom)
Finitely presented group < a, b | a^3, b^4, a^-1*b*a*b^-1 >
>>> D = C3.semidirect_product(C4, hom, reduced=True); D
Finitely presented group < a, b | a^3, b^4, a^-1*b*a*b^-1 >
>>> D.as_permutation_group().is_cyclic()
True

You can turn off the checks for the validity of the input morphisms. This check is expensive but behavior is unpredictable if inputs are invalid and are not caught by these tests:

sage: C5 = groups.presentation.Cyclic(5)
sage: C12 = groups.presentation.Cyclic(12)
sage: hom = (C5.gens(), [(C12.gens(), C12.gens())])
sage: sp = C5.semidirect_product(C12, hom, check=False); sp
Finitely presented group < a, b | a^5, b^12, a^-1*b*a*b^-1 >
sage: sp.as_permutation_group().is_cyclic(), sp.order()
(True, 60)

>>> from sage.all import *
>>> C5 = groups.presentation.Cyclic(Integer(5))
>>> C12 = groups.presentation.Cyclic(Integer(12))
>>> hom = (C5.gens(), [(C12.gens(), C12.gens())])
>>> sp = C5.semidirect_product(C12, hom, check=False); sp
Finitely presented group < a, b | a^5, b^12, a^-1*b*a*b^-1 >
>>> sp.as_permutation_group().is_cyclic(), sp.order()
(True, 60)
simplification_isomorphism()[source]#

Return an isomorphism from self to a finitely presented group with a (hopefully) simpler presentation.

EXAMPLES:

sage: G.<a,b,c> = FreeGroup()
sage: H = G / [a*b*c, a*b^2, c*b/c^2]
sage: I = H.simplification_isomorphism()
sage: I
Generic morphism:
  From: Finitely presented group < a, b, c | a*b*c, a*b^2, c*b*c^-2 >
  To:   Finitely presented group < b |  >
  Defn: a |--> b^-2
        b |--> b
        c |--> b
sage: I(a)
b^-2
sage: I(b)
b
sage: I(c)
b

>>> from sage.all import *
>>> G = FreeGroup(names=('a', 'b', 'c',)); (a, b, c,) = G._first_ngens(3)
>>> H = G / [a*b*c, a*b**Integer(2), c*b/c**Integer(2)]
>>> I = H.simplification_isomorphism()
>>> I
Generic morphism:
  From: Finitely presented group < a, b, c | a*b*c, a*b^2, c*b*c^-2 >
  To:   Finitely presented group < b |  >
  Defn: a |--> b^-2
        b |--> b
        c |--> b
>>> I(a)
b^-2
>>> I(b)
b
>>> I(c)
b

ALGORITHM:

Uses GAP.

simplified()[source]#

Return an isomorphic group with a (hopefully) simpler presentation.

OUTPUT:

A new finitely presented group. Use simplification_isomorphism() if you want to know the isomorphism.

EXAMPLES:

sage: G.<x,y> = FreeGroup()
sage: H = G /  [x ^5, y ^4, y*x*y^3*x ^3]
sage: H
Finitely presented group < x, y | x^5, y^4, y*x*y^3*x^3 >
sage: H.simplified()
Finitely presented group < x, y | y^4, y*x*y^-1*x^-2, x^5 >

>>> from sage.all import *
>>> G = FreeGroup(names=('x', 'y',)); (x, y,) = G._first_ngens(2)
>>> H = G /  [x **Integer(5), y **Integer(4), y*x*y**Integer(3)*x **Integer(3)]
>>> H
Finitely presented group < x, y | x^5, y^4, y*x*y^3*x^3 >
>>> H.simplified()
Finitely presented group < x, y | y^4, y*x*y^-1*x^-2, x^5 >

A more complicate example:

sage: G.<e0, e1, e2, e3, e4, e5, e6, e7, e8, e9> = FreeGroup()
sage: rels = [e6, e5, e3, e9, e4*e7^-1*e6, e9*e7^-1*e0,
....:         e0*e1^-1*e2, e5*e1^-1*e8, e4*e3^-1*e8, e2]
sage: H = G.quotient(rels);  H
Finitely presented group < e0, e1, e2, e3, e4, e5, e6, e7, e8, e9 |
e6, e5, e3, e9, e4*e7^-1*e6, e9*e7^-1*e0, e0*e1^-1*e2, e5*e1^-1*e8, e4*e3^-1*e8, e2 >
sage: H.simplified()
Finitely presented group < e0 | e0^2 >

>>> from sage.all import *
>>> G = FreeGroup(names=('e0', 'e1', 'e2', 'e3', 'e4', 'e5', 'e6', 'e7', 'e8', 'e9',)); (e0, e1, e2, e3, e4, e5, e6, e7, e8, e9,) = G._first_ngens(10)
>>> rels = [e6, e5, e3, e9, e4*e7**-Integer(1)*e6, e9*e7**-Integer(1)*e0,
...         e0*e1**-Integer(1)*e2, e5*e1**-Integer(1)*e8, e4*e3**-Integer(1)*e8, e2]
>>> H = G.quotient(rels);  H
Finitely presented group < e0, e1, e2, e3, e4, e5, e6, e7, e8, e9 |
e6, e5, e3, e9, e4*e7^-1*e6, e9*e7^-1*e0, e0*e1^-1*e2, e5*e1^-1*e8, e4*e3^-1*e8, e2 >
>>> H.simplified()
Finitely presented group < e0 | e0^2 >
sorted_presentation()[source]#

Return the same presentation with the relations sorted to ensure equality.

OUTPUT: a new finitely presented group with the relations sorted

EXAMPLES:

sage: G = FreeGroup(2) / [(1, 2, -1, -2), ()]; G
Finitely presented group < x0, x1 | x0*x1*x0^-1*x1^-1, 1 >
sage: G.sorted_presentation()
Finitely presented group < x0, x1 | 1, x1^-1*x0^-1*x1*x0 >

>>> from sage.all import *
>>> G = FreeGroup(Integer(2)) / [(Integer(1), Integer(2), -Integer(1), -Integer(2)), ()]; G
Finitely presented group < x0, x1 | x0*x1*x0^-1*x1^-1, 1 >
>>> G.sorted_presentation()
Finitely presented group < x0, x1 | 1, x1^-1*x0^-1*x1*x0 >
structure_description(G, latex=False)[source]#

Return a string that tries to describe the structure of G.

This methods wraps GAP’s StructureDescription method.

For full details, including the form of the returned string and the algorithm to build it, see GAP’s documentation.

INPUT:

  • latex – a boolean (default: False); if True, return a LaTeX formatted string

OUTPUT: string

Warning

From GAP’s documentation: The string returned by StructureDescription is not an isomorphism invariant: non-isomorphic groups can have the same string value, and two isomorphic groups in different representations can produce different strings.

EXAMPLES:

sage: # needs sage.groups
sage: G = CyclicPermutationGroup(6)
sage: G.structure_description()
'C6'
sage: G.structure_description(latex=True)
'C_{6}'
sage: G2 = G.direct_product(G, maps=False)
sage: LatexExpr(G2.structure_description(latex=True))
C_{6} \times C_{6}

>>> from sage.all import *
>>> # needs sage.groups
>>> G = CyclicPermutationGroup(Integer(6))
>>> G.structure_description()
'C6'
>>> G.structure_description(latex=True)
'C_{6}'
>>> G2 = G.direct_product(G, maps=False)
>>> LatexExpr(G2.structure_description(latex=True))
C_{6} \times C_{6}

This method is mainly intended for small groups or groups with few normal subgroups. Even then there are some surprises:

sage: D3 = DihedralGroup(3)                                                     # needs sage.groups
sage: D3.structure_description()                                                # needs sage.groups
'S3'

>>> from sage.all import *
>>> D3 = DihedralGroup(Integer(3))                                                     # needs sage.groups
>>> D3.structure_description()                                                # needs sage.groups
'S3'

We use the Sage notation for the degree of dihedral groups:

sage: D4 = DihedralGroup(4)                                                     # needs sage.groups
sage: D4.structure_description()                                                # needs sage.groups
'D4'

>>> from sage.all import *
>>> D4 = DihedralGroup(Integer(4))                                                     # needs sage.groups
>>> D4.structure_description()                                                # needs sage.groups
'D4'

Works for finitely presented groups (Issue #17573):

sage: F.<x, y> = FreeGroup()                                                    # needs sage.groups
sage: G = F / [x^2*y^-1, x^3*y^2, x*y*x^-1*y^-1]                                # needs sage.groups
sage: G.structure_description()                                                 # needs sage.groups
'C7'

>>> from sage.all import *
>>> F = FreeGroup(names=('x', 'y',)); (x, y,) = F._first_ngens(2)# needs sage.groups
>>> G = F / [x**Integer(2)*y**-Integer(1), x**Integer(3)*y**Integer(2), x*y*x**-Integer(1)*y**-Integer(1)]                                # needs sage.groups
>>> G.structure_description()                                                 # needs sage.groups
'C7'

And matrix groups (Issue #17573):

sage: groups.matrix.GL(4,2).structure_description()                             # needs sage.libs.gap sage.modules
'A8'

>>> from sage.all import *
>>> groups.matrix.GL(Integer(4),Integer(2)).structure_description()                             # needs sage.libs.gap sage.modules
'A8'
class sage.groups.finitely_presented.FinitelyPresentedGroupElement(parent, x, check=True)[source]#

Bases: FreeGroupElement

A wrapper of GAP’s Finitely Presented Group elements.

The elements are created by passing the Tietze list that determines them.

EXAMPLES:

sage: G = FreeGroup('a, b')
sage: H = G / [G([1]), G([2, 2, 2])]
sage: H([1, 2, 1, -1])
a*b
sage: H([1, 2, 1, -2])
a*b*a*b^-1
sage: x = H([1, 2, -1, -2])
sage: x
a*b*a^-1*b^-1
sage: y = H([2, 2, 2, 1, -2, -2, -2])
sage: y
b^3*a*b^-3
sage: x*y
a*b*a^-1*b^2*a*b^-3
sage: x^(-1)
b*a*b^-1*a^-1

>>> from sage.all import *
>>> G = FreeGroup('a, b')
>>> H = G / [G([Integer(1)]), G([Integer(2), Integer(2), Integer(2)])]
>>> H([Integer(1), Integer(2), Integer(1), -Integer(1)])
a*b
>>> H([Integer(1), Integer(2), Integer(1), -Integer(2)])
a*b*a*b^-1
>>> x = H([Integer(1), Integer(2), -Integer(1), -Integer(2)])
>>> x
a*b*a^-1*b^-1
>>> y = H([Integer(2), Integer(2), Integer(2), Integer(1), -Integer(2), -Integer(2), -Integer(2)])
>>> y
b^3*a*b^-3
>>> x*y
a*b*a^-1*b^2*a*b^-3
>>> x**(-Integer(1))
b*a*b^-1*a^-1
Tietze()[source]#

Return the Tietze list of the element.

The Tietze list of a word is a list of integers that represent the letters in the word. A positive integer \(i\) represents the letter corresponding to the \(i\)-th generator of the group. Negative integers represent the inverses of generators.

OUTPUT: tuple of integers

EXAMPLES:

sage: G = FreeGroup('a, b')
sage: H = G / (G([1]), G([2, 2, 2]))
sage: H.inject_variables()
Defining a, b
sage: a.Tietze()
(1,)
sage: x = a^2*b^(-3)*a^(-2)
sage: x.Tietze()
(1, 1, -2, -2, -2, -1, -1)

>>> from sage.all import *
>>> G = FreeGroup('a, b')
>>> H = G / (G([Integer(1)]), G([Integer(2), Integer(2), Integer(2)]))
>>> H.inject_variables()
Defining a, b
>>> a.Tietze()
(1,)
>>> x = a**Integer(2)*b**(-Integer(3))*a**(-Integer(2))
>>> x.Tietze()
(1, 1, -2, -2, -2, -1, -1)
class sage.groups.finitely_presented.GroupMorphismWithGensImages[source]#

Bases: SetMorphism

Class used for morphisms from finitely presented groups to other groups. It just adds the images of the generators at the end of the representation.

EXAMPLES:

sage: F = FreeGroup(3)
sage: G = F / [F([1, 2, 3, 1, 2, 3]), F([1, 1, 1])]
sage: H = AlternatingGroup(3)
sage: HS = G.Hom(H)
sage: from sage.groups.finitely_presented import GroupMorphismWithGensImages
sage: GroupMorphismWithGensImages(HS, lambda a: H.one())
Generic morphism:
From: Finitely presented group < x0, x1, x2 | (x0*x1*x2)^2, x0^3 >
To:   Alternating group of order 3!/2 as a permutation group
Defn: x0 |--> ()
      x1 |--> ()
      x2 |--> ()

>>> from sage.all import *
>>> F = FreeGroup(Integer(3))
>>> G = F / [F([Integer(1), Integer(2), Integer(3), Integer(1), Integer(2), Integer(3)]), F([Integer(1), Integer(1), Integer(1)])]
>>> H = AlternatingGroup(Integer(3))
>>> HS = G.Hom(H)
>>> from sage.groups.finitely_presented import GroupMorphismWithGensImages
>>> GroupMorphismWithGensImages(HS, lambda a: H.one())
Generic morphism:
From: Finitely presented group < x0, x1, x2 | (x0*x1*x2)^2, x0^3 >
To:   Alternating group of order 3!/2 as a permutation group
Defn: x0 |--> ()
      x1 |--> ()
      x2 |--> ()
class sage.groups.finitely_presented.RewritingSystem(G)[source]#

Bases: object

A class that wraps GAP’s rewriting systems.

A rewriting system is a set of rules that allow to transform one word in the group to an equivalent one.

If the rewriting system is confluent, then the transformed word is a unique reduced form of the element of the group.

Warning

Note that the process of making a rewriting system confluent might not end.

INPUT:

  • G – a group

REFERENCES:

EXAMPLES:

sage: F.<a,b> = FreeGroup()
sage: G = F / [a*b/a/b]
sage: k = G.rewriting_system()
sage: k
Rewriting system of Finitely presented group < a, b | a*b*a^-1*b^-1 >
with rules:
    a*b*a^-1*b^-1    --->    1

sage: k.reduce(a*b*a*b)
(a*b)^2
sage: k.make_confluent()
sage: k
Rewriting system of Finitely presented group < a, b | a*b*a^-1*b^-1 >
with rules:
    b^-1*a^-1    --->    a^-1*b^-1
    b^-1*a    --->    a*b^-1
    b*a^-1    --->    a^-1*b
    b*a    --->    a*b

sage: k.reduce(a*b*a*b)
a^2*b^2

>>> from sage.all import *
>>> F = FreeGroup(names=('a', 'b',)); (a, b,) = F._first_ngens(2)
>>> G = F / [a*b/a/b]
>>> k = G.rewriting_system()
>>> k
Rewriting system of Finitely presented group < a, b | a*b*a^-1*b^-1 >
with rules:
    a*b*a^-1*b^-1    --->    1

>>> k.reduce(a*b*a*b)
(a*b)^2
>>> k.make_confluent()
>>> k
Rewriting system of Finitely presented group < a, b | a*b*a^-1*b^-1 >
with rules:
    b^-1*a^-1    --->    a^-1*b^-1
    b^-1*a    --->    a*b^-1
    b*a^-1    --->    a^-1*b
    b*a    --->    a*b

>>> k.reduce(a*b*a*b)
a^2*b^2

Todo

  • Include support for different orderings (currently only shortlex is used).

  • Include the GAP package kbmag for more functionalities, including automatic structures and faster compiled functions.

AUTHORS:

  • Miguel Angel Marco Buzunariz (2013-12-16)

finitely_presented_group()[source]#

The finitely presented group where the rewriting system is defined.

EXAMPLES:

sage: F = FreeGroup(3)
sage: G = F / [ [1,2,3], [-1,-2,-3], [1,1], [2,2] ]
sage: k = G.rewriting_system()
sage: k.make_confluent()
sage: k
Rewriting system of Finitely presented group < x0, x1, x2 | x0*x1*x2, x0^-1*x1^-1*x2^-1, x0^2, x1^2 >
with rules:
    x0^-1    --->    x0
    x1^-1    --->    x1
    x2^-1    --->    x2
    x0^2    --->    1
    x0*x1    --->    x2
    x0*x2    --->    x1
    x1*x0    --->    x2
    x1^2    --->    1
    x1*x2    --->    x0
    x2*x0    --->    x1
    x2*x1    --->    x0
    x2^2    --->    1
sage: k.finitely_presented_group()
Finitely presented group < x0, x1, x2 | x0*x1*x2, x0^-1*x1^-1*x2^-1, x0^2, x1^2 >

>>> from sage.all import *
>>> F = FreeGroup(Integer(3))
>>> G = F / [ [Integer(1),Integer(2),Integer(3)], [-Integer(1),-Integer(2),-Integer(3)], [Integer(1),Integer(1)], [Integer(2),Integer(2)] ]
>>> k = G.rewriting_system()
>>> k.make_confluent()
>>> k
Rewriting system of Finitely presented group < x0, x1, x2 | x0*x1*x2, x0^-1*x1^-1*x2^-1, x0^2, x1^2 >
with rules:
    x0^-1    --->    x0
    x1^-1    --->    x1
    x2^-1    --->    x2
    x0^2    --->    1
    x0*x1    --->    x2
    x0*x2    --->    x1
    x1*x0    --->    x2
    x1^2    --->    1
    x1*x2    --->    x0
    x2*x0    --->    x1
    x2*x1    --->    x0
    x2^2    --->    1
>>> k.finitely_presented_group()
Finitely presented group < x0, x1, x2 | x0*x1*x2, x0^-1*x1^-1*x2^-1, x0^2, x1^2 >
free_group()[source]#

The free group after which the rewriting system is defined

EXAMPLES:

sage: F = FreeGroup(3)
sage: G = F / [ [1,2,3], [-1,-2,-3] ]
sage: k = G.rewriting_system()
sage: k.free_group()
Free Group on generators {x0, x1, x2}

>>> from sage.all import *
>>> F = FreeGroup(Integer(3))
>>> G = F / [ [Integer(1),Integer(2),Integer(3)], [-Integer(1),-Integer(2),-Integer(3)] ]
>>> k = G.rewriting_system()
>>> k.free_group()
Free Group on generators {x0, x1, x2}
gap()[source]#

The gap representation of the rewriting system.

EXAMPLES:

sage: F.<a,b> = FreeGroup()
sage: G = F/[a*a,b*b]
sage: k = G.rewriting_system()
sage: k.gap()
Knuth Bendix Rewriting System for Monoid( [ a, A, b, B ] ) with rules
[ [ a*A, <identity ...> ], [ A*a, <identity ...> ],
  [ b*B, <identity ...> ], [ B*b, <identity ...> ],
  [ a^2, <identity ...> ], [ b^2, <identity ...> ] ]

>>> from sage.all import *
>>> F = FreeGroup(names=('a', 'b',)); (a, b,) = F._first_ngens(2)
>>> G = F/[a*a,b*b]
>>> k = G.rewriting_system()
>>> k.gap()
Knuth Bendix Rewriting System for Monoid( [ a, A, b, B ] ) with rules
[ [ a*A, <identity ...> ], [ A*a, <identity ...> ],
  [ b*B, <identity ...> ], [ B*b, <identity ...> ],
  [ a^2, <identity ...> ], [ b^2, <identity ...> ] ]
is_confluent()[source]#

Return True if the system is confluent and False otherwise.

EXAMPLES:

sage: F = FreeGroup(3)
sage: G = F / [F([1,2,1,2,1,3,-1]),F([2,2,2,1,1,2]),F([1,2,3])]
sage: k = G.rewriting_system()
sage: k.is_confluent()
False
sage: k
Rewriting system of Finitely presented group < x0, x1, x2 | (x0*x1)^2*x0*x2*x0^-1, x1^3*x0^2*x1, x0*x1*x2 >
with rules:
    x0*x1*x2    --->    1
    x1^3*x0^2*x1    --->    1
    (x0*x1)^2*x0*x2*x0^-1    --->    1

sage: k.make_confluent()
sage: k.is_confluent()
True
sage: k
Rewriting system of Finitely presented group < x0, x1, x2 | (x0*x1)^2*x0*x2*x0^-1, x1^3*x0^2*x1, x0*x1*x2 >
with rules:
    x0^-1    --->    x0
    x1^-1    --->    x1
    x0^2    --->    1
    x0*x1    --->    x2^-1
    x0*x2^-1    --->    x1
    x1*x0    --->    x2
    x1^2    --->    1
    x1*x2^-1    --->    x0*x2
    x1*x2    --->    x0
    x2^-1*x0    --->    x0*x2
    x2^-1*x1    --->    x0
    x2^-2    --->    x2
    x2*x0    --->    x1
    x2*x1    --->    x0*x2
    x2^2    --->    x2^-1

>>> from sage.all import *
>>> F = FreeGroup(Integer(3))
>>> G = F / [F([Integer(1),Integer(2),Integer(1),Integer(2),Integer(1),Integer(3),-Integer(1)]),F([Integer(2),Integer(2),Integer(2),Integer(1),Integer(1),Integer(2)]),F([Integer(1),Integer(2),Integer(3)])]
>>> k = G.rewriting_system()
>>> k.is_confluent()
False
>>> k
Rewriting system of Finitely presented group < x0, x1, x2 | (x0*x1)^2*x0*x2*x0^-1, x1^3*x0^2*x1, x0*x1*x2 >
with rules:
    x0*x1*x2    --->    1
    x1^3*x0^2*x1    --->    1
    (x0*x1)^2*x0*x2*x0^-1    --->    1

>>> k.make_confluent()
>>> k.is_confluent()
True
>>> k
Rewriting system of Finitely presented group < x0, x1, x2 | (x0*x1)^2*x0*x2*x0^-1, x1^3*x0^2*x1, x0*x1*x2 >
with rules:
    x0^-1    --->    x0
    x1^-1    --->    x1
    x0^2    --->    1
    x0*x1    --->    x2^-1
    x0*x2^-1    --->    x1
    x1*x0    --->    x2
    x1^2    --->    1
    x1*x2^-1    --->    x0*x2
    x1*x2    --->    x0
    x2^-1*x0    --->    x0*x2
    x2^-1*x1    --->    x0
    x2^-2    --->    x2
    x2*x0    --->    x1
    x2*x1    --->    x0*x2
    x2^2    --->    x2^-1
make_confluent()[source]#

Apply the Knuth-Bendix algorithm to try to transform the rewriting system into a confluent one.

Note that this method does not return any object, just changes the rewriting system internally.

Warning

This algorithm is not granted to finish. Although it may be useful in some occasions to run it, interrupt it manually after some time and use then the transformed rewriting system. Even if it is not confluent, it could be used to reduce some words.

ALGORITHM:

Uses GAP’s MakeConfluent.

EXAMPLES:

sage: F.<a,b> = FreeGroup()
sage: G = F / [a^2,b^3,(a*b/a)^3,b*a*b*a]
sage: k = G.rewriting_system()
sage: k
Rewriting system of Finitely presented group < a, b | a^2, b^3, a*b^3*a^-1, (b*a)^2 >
with rules:
    a^2    --->    1
    b^3    --->    1
    (b*a)^2    --->    1
    a*b^3*a^-1    --->    1

sage: k.make_confluent()
sage: k
Rewriting system of Finitely presented group < a, b | a^2, b^3, a*b^3*a^-1, (b*a)^2 >
with rules:
    a^-1    --->    a
    a^2    --->    1
    b^-1*a    --->    a*b
    b^-2    --->    b
    b*a    --->    a*b^-1
    b^2    --->    b^-1

>>> from sage.all import *
>>> F = FreeGroup(names=('a', 'b',)); (a, b,) = F._first_ngens(2)
>>> G = F / [a**Integer(2),b**Integer(3),(a*b/a)**Integer(3),b*a*b*a]
>>> k = G.rewriting_system()
>>> k
Rewriting system of Finitely presented group < a, b | a^2, b^3, a*b^3*a^-1, (b*a)^2 >
with rules:
    a^2    --->    1
    b^3    --->    1
    (b*a)^2    --->    1
    a*b^3*a^-1    --->    1

>>> k.make_confluent()
>>> k
Rewriting system of Finitely presented group < a, b | a^2, b^3, a*b^3*a^-1, (b*a)^2 >
with rules:
    a^-1    --->    a
    a^2    --->    1
    b^-1*a    --->    a*b
    b^-2    --->    b
    b*a    --->    a*b^-1
    b^2    --->    b^-1
reduce(element)[source]#

Apply the rules in the rewriting system to the element, to obtain a reduced form.

If the rewriting system is confluent, this reduced form is unique for all words representing the same element.

EXAMPLES:

sage: F.<a,b> = FreeGroup()
sage: G = F/[a^2, b^3, (a*b/a)^3, b*a*b*a]
sage: k = G.rewriting_system()
sage: k.reduce(b^4)
b
sage: k.reduce(a*b*a)
a*b*a

>>> from sage.all import *
>>> F = FreeGroup(names=('a', 'b',)); (a, b,) = F._first_ngens(2)
>>> G = F/[a**Integer(2), b**Integer(3), (a*b/a)**Integer(3), b*a*b*a]
>>> k = G.rewriting_system()
>>> k.reduce(b**Integer(4))
b
>>> k.reduce(a*b*a)
a*b*a
rules()[source]#

Return the rules that form the rewriting system.

OUTPUT:

A dictionary containing the rules of the rewriting system. Each key is a word in the free group, and its corresponding value is the word to which it is reduced.

EXAMPLES:

sage: F.<a,b> = FreeGroup()
sage: G = F / [a*a*a,b*b*a*a]
sage: k = G.rewriting_system()
sage: k
Rewriting system of Finitely presented group < a, b | a^3, b^2*a^2 >
with rules:
    a^3    --->    1
    b^2*a^2    --->    1

sage: k.rules()
{a^3: 1, b^2*a^2: 1}
sage: k.make_confluent()
sage: sorted(k.rules().items())
[(a^-2, a), (a^-1*b^-1, a*b), (a^-1*b, b^-1), (a^2, a^-1),
 (a*b^-1, b), (b^-1*a^-1, a*b), (b^-1*a, b), (b^-2, a^-1),
 (b*a^-1, b^-1), (b*a, a*b), (b^2, a)]

>>> from sage.all import *
>>> F = FreeGroup(names=('a', 'b',)); (a, b,) = F._first_ngens(2)
>>> G = F / [a*a*a,b*b*a*a]
>>> k = G.rewriting_system()
>>> k
Rewriting system of Finitely presented group < a, b | a^3, b^2*a^2 >
with rules:
    a^3    --->    1
    b^2*a^2    --->    1

>>> k.rules()
{a^3: 1, b^2*a^2: 1}
>>> k.make_confluent()
>>> sorted(k.rules().items())
[(a^-2, a), (a^-1*b^-1, a*b), (a^-1*b, b^-1), (a^2, a^-1),
 (a*b^-1, b), (b^-1*a^-1, a*b), (b^-1*a, b), (b^-2, a^-1),
 (b*a^-1, b^-1), (b*a, a*b), (b^2, a)]
sage.groups.finitely_presented.wrap_FpGroup(libgap_fpgroup)[source]#

Wrap a GAP finitely presented group.

This function changes the comparison method of libgap_free_group to comparison by Python id. If you want to put the LibGAP free group into a container (set, dict) then you should understand the implications of _set_compare_by_id(). To be safe, it is recommended that you just work with the resulting Sage FinitelyPresentedGroup.

INPUT:

  • libgap_fpgroup – a LibGAP finitely presented group

OUTPUT: a Sage FinitelyPresentedGroup

EXAMPLES:

First construct a LibGAP finitely presented group:

sage: F = libgap.FreeGroup(['a', 'b'])
sage: a_cubed = F.GeneratorsOfGroup()[0] ^ 3
sage: P = F / libgap([ a_cubed ]);   P
<fp group of size infinity on the generators [ a, b ]>
sage: type(P)
<class 'sage.libs.gap.element.GapElement'>

>>> from sage.all import *
>>> F = libgap.FreeGroup(['a', 'b'])
>>> a_cubed = F.GeneratorsOfGroup()[Integer(0)] ** Integer(3)
>>> P = F / libgap([ a_cubed ]);   P
<fp group of size infinity on the generators [ a, b ]>
>>> type(P)
<class 'sage.libs.gap.element.GapElement'>

Now wrap it:

sage: from sage.groups.finitely_presented import wrap_FpGroup
sage: wrap_FpGroup(P)
Finitely presented group < a, b | a^3 >

>>> from sage.all import *
>>> from sage.groups.finitely_presented import wrap_FpGroup
>>> wrap_FpGroup(P)
Finitely presented group < a, b | a^3 >