Free Groups#

Free groups and finitely presented groups are implemented as a wrapper over the corresponding GAP objects.

A free group can be created by giving the number of generators, or their names. It is also possible to create indexed generators:

sage: G.<x,y,z> = FreeGroup();  G
Free Group on generators {x, y, z}
sage: FreeGroup(3)
Free Group on generators {x0, x1, x2}
sage: FreeGroup('a,b,c')
Free Group on generators {a, b, c}
sage: FreeGroup(3,'t')
Free Group on generators {t0, t1, t2}

The elements can be created by operating with the generators, or by passing a list with the indices of the letters to the group:


sage: G.<a,b,c> = FreeGroup()
sage: a*b*c*a
sage: G([1,2,3,1])
sage: a * b / c * b^2
sage: G([1,1,2,-1,-3,2])

You can use call syntax to replace the generators with a set of arbitrary ring elements:

sage: g =  a * b / c * b^2
sage: g(1,2,3)
sage: M1 = identity_matrix(2)
sage: M2 = matrix([[1,1],[0,1]])
sage: M3 = matrix([[0,1],[1,0]])
sage: g([M1, M2, M3])
[1 3]
[1 2]


  • Miguel Angel Marco Buzunariz

  • Volker Braun

sage.groups.free_group.FreeGroup(n=None, names='x', index_set=None, abelian=False, **kwds)#

Construct a Free Group.


  • n – integer or None (default). The number of generators. If not specified the names are counted.

  • names – string or list/tuple/iterable of strings (default: 'x'). The generator names or name prefix.

  • index_set – (optional) an index set for the generators; if specified then the optional keyword abelian can be used

  • abelian – (default: False) whether to construct a free abelian group or a free group


If you want to create a free group, it is currently preferential to use Groups().free(...) as that does not load GAP.


sage: G.<a,b> = FreeGroup();  G
Free Group on generators {a, b}
sage: H = FreeGroup('a, b')
sage: G is H
sage: FreeGroup(0)
Free Group on generators {}

The entry can be either a string with the names of the generators, or the number of generators and the prefix of the names to be given. The default prefix is 'x'

sage: FreeGroup(3)
Free Group on generators {x0, x1, x2}
sage: FreeGroup(3, 'g')
Free Group on generators {g0, g1, g2}
sage: FreeGroup()
Free Group on generators {x}

We give two examples using the index_set option:

sage: FreeGroup(index_set=ZZ)                                                   # needs sage.combinat
Free group indexed by Integer Ring
sage: FreeGroup(index_set=ZZ, abelian=True)                                     # needs sage.combinat
Free abelian group indexed by Integer Ring
class sage.groups.free_group.FreeGroupElement(parent, x)#

Bases: ElementLibGAP

A wrapper of GAP’s Free Group elements.


  • x – something that determines the group element. Either a GapElement or the Tietze list (see Tietze()) of the group element.

  • parent – the parent FreeGroup.


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

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.


A tuple of integers.


sage: G.<a,b> = FreeGroup()
sage: a.Tietze()
sage: x = a^2 * b^(-3) * a^(-2)
sage: x.Tietze()
(1, 1, -2, -2, -2, -1, -1)
fox_derivative(gen, im_gens=None, ring=None)#

Return the Fox derivative of self with respect to a given generator gen of the free group.

Let \(F\) be a free group with free generators \(x_1, x_2, \ldots, x_n\). Let \(j \in \left\{ 1, 2, \ldots , n \right\}\). Let \(a_1, a_2, \ldots, a_n\) be \(n\) invertible elements of a ring \(A\). Let \(a : F \to A^\times\) be the (unique) homomorphism from \(F\) to the multiplicative group of invertible elements of \(A\) which sends each \(x_i\) to \(a_i\). Then, we can define a map \(\partial_j : F \to A\) by the requirements that

\[\partial_j (x_i) = \delta_{i, j} \qquad \qquad \text{ for all indices } i \text{ and } j\]


\[\partial_j (uv) = \partial_j(u) + a(u) \partial_j(v) \qquad \qquad \text{ for all } u, v \in F .\]

This map \(\partial_j\) is called the \(j\)-th Fox derivative on \(F\) induced by \((a_1, a_2, \ldots, a_n)\).

The most well-known case is when \(A\) is the group ring \(\ZZ [F]\) of \(F\) over \(\ZZ\), and when \(a_i = x_i \in A\). In this case, \(\partial_j\) is simply called the \(j\)-th Fox derivative on \(F\).


  • gen – the generator with respect to which the derivative will be computed. If this is \(x_j\), then the method will return \(\partial_j\).

  • im_gens (optional) – the images of the generators (given as a list or iterable). This is the list \((a_1, a_2, \ldots, a_n)\). If not provided, it defaults to \((x_1, x_2, \ldots, x_n)\) in the group ring \(\ZZ [F]\).

  • ring (optional) – the ring in which the elements of the list \((a_1, a_2, \ldots, a_n)\) lie. If not provided, this ring is inferred from these elements.


The fox derivative of self with respect to gen (induced by im_gens). By default, it is an element of the group algebra with integer coefficients. If im_gens are provided, the result lives in the algebra where im_gens live.


sage: G = FreeGroup(5)
sage: G.inject_variables()
Defining x0, x1, x2, x3, x4
sage: (~x0*x1*x0*x2*~x0).fox_derivative(x0)
-x0^-1 + x0^-1*x1 - x0^-1*x1*x0*x2*x0^-1
sage: (~x0*x1*x0*x2*~x0).fox_derivative(x1)
sage: (~x0*x1*x0*x2*~x0).fox_derivative(x2)
sage: (~x0*x1*x0*x2*~x0).fox_derivative(x3)

If im_gens is given, the images of the generators are mapped to them:

sage: F = FreeGroup(3)
sage: a = F([2,1,3,-1,2])
sage: a.fox_derivative(F([1]))
x1 - x1*x0*x2*x0^-1
sage: R.<t> = LaurentPolynomialRing(ZZ)
sage: a.fox_derivative(F([1]),[t,t,t])
t - t^2
sage: S.<t1,t2,t3> = LaurentPolynomialRing(ZZ)
sage: a.fox_derivative(F([1]),[t1,t2,t3])
-t2*t3 + t2
sage: R.<x,y,z> = QQ[]
sage: a.fox_derivative(F([1]),[x,y,z])
-y*z + y
sage: a.inverse().fox_derivative(F([1]),[x,y,z])
(z - 1)/(y*z)

The optional parameter ring determines the ring \(A\):

sage: u = a.fox_derivative(F([1]), [1,2,3], ring=QQ)
sage: u
sage: parent(u)
Rational Field
sage: u = a.fox_derivative(F([1]), [1,2,3], ring=R)
sage: u
sage: parent(u)
Multivariate Polynomial Ring in x, y, z over Rational Field

Return the syllables of the word.

Consider a free group element \(g = x_1^{n_1} x_2^{n_2} \cdots x_k^{n_k}\). The uniquely-determined subwords \(x_i^{e_i}\) consisting only of powers of a single generator are called the syllables of \(g\).


The tuple of syllables. Each syllable is given as a pair \((x_i, e_i)\) consisting of a generator and a non-zero integer.


sage: G.<a,b> = FreeGroup()
sage: w = a^2 * b^-1 * a^3
sage: w.syllables()
((a, 2), (b, -1), (a, 3))
class sage.groups.free_group.FreeGroup_class(generator_names, libgap_free_group=None)#

Bases: UniqueRepresentation, Group, ParentLibGAP

A class that wraps GAP’s FreeGroup

See FreeGroup() for details.


alias of FreeGroupElement


Return the Abelian invariants of self.

The Abelian invariants are given by a list of integers \(i_1 \dots i_j\), such that the abelianization of the group is isomorphic to

\[\ZZ / (i_1) \times \dots \times \ZZ / (i_j)\]


sage: F.<a,b> = FreeGroup()
sage: F.abelian_invariants()
(0, 0)
quotient(relations, **kwds)#

Return the quotient of self by the normal subgroup generated by the given elements.

This quotient is a finitely presented groups with the same generators as self, and relations given by the elements of relations.


  • relations – A list/tuple/iterable with the elements of the free group.

  • further named arguments, that are passed to the constructor of a finitely presented group.


A finitely presented group, with generators corresponding to the generators of the free group, and relations corresponding to the elements in relations.


sage: F.<a,b> = FreeGroup()
sage: F.quotient([a*b^2*a, b^3])
Finitely presented group < a, b | a*b^2*a, b^3 >

Division is shorthand for quotient()

sage: F /  [a*b^2*a, b^3]
Finitely presented group < a, b | a*b^2*a, b^3 >

Relations are converted to the free group, even if they are not elements of it (if possible)

sage: F1.<a,b,c,d> = FreeGroup()
sage: F2.<a,b> = FreeGroup()
sage: r = a*b/a
sage: r.parent()
Free Group on generators {a, b}
sage: F1/[r]
Finitely presented group < a, b, c, d | a*b*a^-1 >

Return the number of generators of self.

Alias for ngens().




sage: G = FreeGroup('a, b');  G
Free Group on generators {a, b}
sage: G.rank()
sage: H = FreeGroup(3, 'x')
sage: H
Free Group on generators {x0, x1, x2}
sage: H.rank()

Test whether x is a FreeGroup_class.


  • x – anything.




sage: from sage.groups.free_group import is_FreeGroup
sage: is_FreeGroup('a string')                                                  # needs sage.combinat
sage: is_FreeGroup(FreeGroup(0))
sage: is_FreeGroup(FreeGroup(index_set=ZZ))                                     # needs sage.combinat

Wrap a LibGAP free 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 FreeGroup_class.


  • libgap_free_group – a LibGAP free group.


A Sage FreeGroup_class.


First construct a LibGAP free group:

sage: F = libgap.FreeGroup(['a', 'b'])
sage: type(F)
<class ''>

Now wrap it:

sage: from sage.groups.free_group import wrap_FreeGroup
sage: wrap_FreeGroup(F)
Free Group on generators {a, b}