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:

EXAMPLES:

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

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)
8/3
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]

AUTHORS:

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.

INPUT:

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

Note

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

EXAMPLES:

sage: G.<a,b> = FreeGroup();  G
Free Group on generators {a, b}
sage: H = FreeGroup('a, b')
sage: G is H
True
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)
Free group indexed by Integer Ring
sage: FreeGroup(index_set=ZZ, abelian=True)
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.

INPUT:

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

EXAMPLES:

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

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.<a,b> = FreeGroup()
sage: a.Tietze()
(1,)
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 $x_{1}, x_{2}, \dots, x_{n}$ . Let $j \in {1, 2, \dots, n}$ . Let $a_{1}, a_{2}, \dots, a_{n}$ 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}) = δ_{i, j} for all indices i and j

and

\partial_{j} (u v) = \partial_{j} (u) + a (u) \partial_{j} (v) for all u, v \in F .

This map $\partial_{j}$ is called the $j$ -th Fox derivative on $F$ induced by $(a_{1}, a_{2}, \dots, a_{n})$ .

The most well-known case is when $A$ is the group ring $Z [F]$ of $F$ over $Z$ , and when $a_{i} = x_{i} \in A$ . In this case, $\partial_{j}$ is simply called the $j$ -th Fox derivative on $F$ .

INPUT:

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 $(a_{1}, a_{2}, \dots, a_{n})$ . If not provided, it defaults to $(x_{1}, x_{2}, \dots, x_{n})$ in the group ring $Z [F]$ .
ring (optional) – the ring in which the elements of the list $(a_{1}, a_{2}, \dots, a_{n})$ lie. If not provided, this ring is inferred from these elements.

OUTPUT:

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.

EXAMPLES:

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)
x0^-1
sage: (~x0*x1*x0*x2*~x0).fox_derivative(x2)
x0^-1*x1*x0
sage: (~x0*x1*x0*x2*~x0).fox_derivative(x3)
0

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
-4
sage: parent(u)
Rational Field
sage: u = a.fox_derivative(F([1]), [1,2,3], ring=R)
sage: u
-4
sage: parent(u)
Multivariate Polynomial Ring in x, y, z over Rational Field

syllables()#

Return the syllables of the word.

Consider a free group element $g = x_{1}^{n_{1}} x_{2}^{n_{2}} \dots x_{k}^{n_{k}}$ . The uniquely-determined subwords $x_{i}^{e_{i}}$ consisting only of powers of a single generator are called the syllables of $g$ .

OUTPUT:

The tuple of syllables. Each syllable is given as a pair $(x_{i}, e_{i})$ consisting of a generator and a non-zero integer.

EXAMPLES:

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.

Element#: alias of FreeGroupElement

abelian_invariants()#

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

Z / (i_{1}) \times \dots \times Z / (i_{j})

EXAMPLES:

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.

INPUT:

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.

OUTPUT:

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

EXAMPLES:

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 >

rank()#

Return the number of generators of self.

Alias for ngens().

OUTPUT:

Integer.

EXAMPLES:

sage: G = FreeGroup('a, b');  G
Free Group on generators {a, b}
sage: G.rank()
2
sage: H = FreeGroup(3, 'x')
sage: H
Free Group on generators {x0, x1, x2}
sage: H.rank()
3

sage.groups.free_group.is_FreeGroup(x)#

Test whether x is a FreeGroup_class.

INPUT:

x – anything.

OUTPUT:

Boolean.

EXAMPLES:

sage: from sage.groups.free_group import is_FreeGroup
sage: is_FreeGroup('a string')
False
sage: is_FreeGroup(FreeGroup(0))
True
sage: is_FreeGroup(FreeGroup(index_set=ZZ))
True

sage.groups.free_group.wrap_FreeGroup(libgap_free_group)#

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.

INPUT:

libgap_free_group – a LibGAP free group.

OUTPUT:

A Sage FreeGroup_class.

EXAMPLES:

First construct a LibGAP free group:

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

Now wrap it:

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