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 >
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 >
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
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 ]>
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>) ]
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'>
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
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)#
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'>
- Element#
alias of
FinitelyPresentedGroupElement
- abelian_alexander_matrix(ring=Rational Field, simplified=True)#
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 algebrasimplified
– boolean (default:False
); if set toTrue
use Gauss elimination and erase rows and columns
OUTPUT:
A
– a matrix with coefficients inR
ideal
– an list of generators of an idealI
ofR = A.base_ring()
such thatR/I
is the group algebra of the abelianization ofself
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
- abelian_invariants()#
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)
ALGORITHM:
Uses GAP.
- abelianization_map()#
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 < | >
- abelianization_to_algebra(ring=Rational Field)#
Return the group algebra of the abelianization of
self
together with the monomials representing the generators ofself
.INPUT:
ring
– (default:QQ
); the base ring for the group algebra ofself
OUTPUT:
ab
– the abelianization ofself
as a finitely presented group with a minimal number \(n\) of generators.R
– a Laurent polynomial ring with \(n\) variables with base ringring
.ideal
– a list of generators of an idealI
inR
such thatR/I
is the group algebra of the abelianization overring
image
– a list with the images of the generators ofself
inR/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, [], [])
- alexander_matrix(im_gens=None)#
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]
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]
- as_permutation_group(limit=4096000)#
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 givenlimit
, aValueError
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?
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)#
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
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)#
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 algebragroebner
– boolean (default:False
); If set toTrue
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
isFalse
the values are the ideals defining the characteristic varieties. If it isTrue
, 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,)]}
- direct_product(H, reduced=False, new_names=True)#
Return the direct product of
self
with finitely presented groupH
.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 groupreduced
– (default:False
) boolean; ifTrue
, then attempt to reduce the presentation of the product groupnew_names
– (default:True
) boolean; IfTrue
, then lexicographical variable names are assigned to the generators of the group to be returned. IfFalse
, 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
withH
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)
We can keep the variable names from
self
andH
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 >
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 < | >
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
AUTHORS:
Davis Shurbert (2013-07-20): initial version
- epimorphisms(H)#
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)]
ALGORITHM:
Uses libgap’s GQuotients function.
- free_group()#
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
- order(limit=4096000)#
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
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()#
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
- rewriting_system()#
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
- semidirect_product(H, hom, check=True, reduced=False)#
The semidirect product of
self
withH
viahom
.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 byself
and can be naturally embedded as a normal subgroup of the semidirect product.hom
– Homomorphism fromself
to the automorphism group ofH
. Given as a pair, with generators ofself
in the first slot and the images of the corresponding generators in the second. These images must be automorphisms ofH
, given again as a pair of generators and images.check
– Boolean (defaultTrue
). IfFalse
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 (defaultFalse
). IfTrue
then the method attempts to reduce the presentation of the output group.
OUTPUT:
The semidirect product of
self
withH
viahom
as a finitely presented group. SeePermutationGroup_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
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
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
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)
- simplification_isomorphism()#
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
ALGORITHM:
Uses GAP.
- simplified()#
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 >
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 >
- sorted_presentation()#
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 >
- structure_description(G, latex=False)#
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
). IfTrue
, 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}
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'
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'
Works for finitely presented groups (github 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'
And matrix groups (github issue #17573):
sage: groups.matrix.GL(4,2).structure_description() # needs sage.libs.gap sage.modules 'A8'
- class sage.groups.finitely_presented.FinitelyPresentedGroupElement(parent, x, check=True)#
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
- Tietze()#
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:
A 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)
- class sage.groups.finitely_presented.GroupMorphismWithGensImages#
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 |--> ()
- class sage.groups.finitely_presented.RewritingSystem(G)#
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
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()#
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 >
- free_group()#
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}
- gap()#
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 ...> ] ]
- is_confluent()#
Return
True
if the system is confluent andFalse
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
- make_confluent()#
Applies 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
- reduce(element)#
Applies 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
- rules()#
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)]
- sage.groups.finitely_presented.wrap_FpGroup(libgap_fpgroup)#
Wrap a GAP finitely presented group.
This function changes the comparison method of
libgap_free_group
to comparison by Pythonid
. 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 SageFinitelyPresentedGroup
.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'>
Now wrap it:
sage: from sage.groups.finitely_presented import wrap_FpGroup sage: wrap_FpGroup(P) Finitely presented group < a, b | a^3 >