Finitely Presented Groups

Introduction

This module presents the functionality designed for computing with finitely- presented groups (fp-groups for short). The name of the corresponding SymPy object is FpGroup. The functions or classes described here are studied under computational group theory. All code examples assume:

>>> from sympy.combinatorics.free_groups import free_group, vfree_group, xfree_group
>>> from sympy.combinatorics.fp_groups import FpGroup, CosetTable, coset_enumeration_r

Overview of Facilities

The facilities provided for fp-groups fall into a number of natural groupings

  • The construction of fp-groups using a free group and a list of words in generators of that free group.
  • Index determination using the famous Todd-Coxeter procedure.
  • The construction of all subgroups having index less than some (small) specified positive integer, using the Low-Index Subgroups algorithm.
  • Algorithms for computing presentations of a subgroup of finite index in a group defined by finite presentation.

For a description of fundamental algorithms of finitely presented groups we often make use of Handbook of Computational Group Theory.

The Construction of Finitely Presented Groups

Finitely presented groups are constructed by factoring a free group by a set of relators. The set of relators is taken in as a list of words in generators of free group in SymPy, using a list provides ordering to the relators. If the list of relators is empty, the associated free group is returned.

Example of construction of a finitely-presented group. The symmetric group of degree 4 may be represented as a two generator group with presentation \(\langle a, b \mid a^2, b^3, (ab)^4 \rangle\). Giving the relations as a list of relators, group in SymPy would be specified as:

>>> F, a, b = free_group("a, b")
>>> G = FpGroup(F, [a**2, b**3, (a*b)**4])
>>> G
<fp group on the generators (a, b)>

Currently groups with relators having presentation like \(\langle r, s, t \mid r^2, s^2, t^2, rst = str = trs \rangle\) will have to be specified as:

>>> F, r, s, t = free_group("r, s, t")
>>> G = FpGroup(F, [r**2, s**2, t**2, r*s*t*r**-1*t**-1*s**-1, s*t*r*s**-1*r**-1*t**-1])

Obviously this is not a unique way to make that particular group, but the point is that in case of equality with non-identity the user has to manually do that.

Free Groups and Words

Construction of a Free Group

free_group("gen0, gen1, ..., gen_(n-1)") constructs a free group F on n generators, where n is a positive integer. The \(i\)-th generator of \(F\) may be obtained using the method .generators[i], \(i = 0, \ldots n-1\).

>>> F, x, y = free_group("x, y")

creates a free group F of rank 2 and assigns the variables x and y to the two generators.

>>> F = vfree_group("x, y")
>>> F
<free group on the generators (x, y)>

creates a free group F of rank 2, with tuple of generators F.generators, and inserts x and y as generators into the global namespace.

>>> F = xfree_group("x, y")
>>> F
(<free group on the generators (x, y)>, (x, y))
>>> x**2
x**2

creates a free groups F[0] of rank 2, with tuple of generators F[1].

Construction of words

This section is applicable to words of FreeGroup as well as FpGroup. When we say word in SymPy, it actually means a reduced word , since the words are automatically reduced. Given a group G defined on \(n\) generators \(x_1, x_2, x_3, \ldots, x_n\), a word is constructed as \(s_1^{r_1}s_2^{r_2} \cdots s_k^{r_k}\) where \(s_i \in \{x_1, x_2, \ldots, x_n\}\) , \(r_i \in \mathbb{Z}\) for all \(k\).

Each word can be constructed in a variety of ways, since after reduction they may be equivalent.

Coset Enumeration: The Todd-Coxeter Algorithm

This section describes the use of coset enumeration techniques in SymPy. The algorithm used for coset enumeration procedure is Todd-Coxeter algorithm and is developed in Sympy using [Ho05] and [CDHW73]. The reader should consult [CDHW73] and [Hav91] for a general description of the algorithm.

We have two strategies of coset enumeration relator-based and coset-table based and the two have been implemented as coset_enumeration_r, coset_enumeration_c respectively. The two strategies differ in the way they make new definitions for the cosets.

Though from the user point of view it is suggested to rather use the .coset_enumeration method of FpGroup and specify the strategy argument.

strategy:
(default=”relator_based”) specifies the strategy of coset enumeration to be used, possible values are “relator_based” or “coset_table_based”.

CosetTable

Class used to manipulate the information regarding the coset enumeration of the finitely presented group G on the cosets of the subgroup H.

Basically a coset table CosetTable(G,H), is the permutation representation of the finitely presented group on the cosets of a subgroup. Most of the set theoretic and group functions use the regular representation of G, i.e., the coset table of G over the trivial subgroup.

The actual mathematical coset table is obtained using .table attribute and is a list of lists. For each generator g of G it contains a column and the next column corresponds to g**-1 and so on for other generators, so in total it has 2*G.rank() columns. Each column is simply a list of integers. If l is the generator list for the generator \(g\) and if l[i] = j then generator g takes the coset \(i\) to the coset \(j\) by multiplication from the right.

For finitely presented groups, a coset table is computed by a Todd-Coxeter coset enumeration. Note that you may influence the performance of that enumeration by changing the values of the variable CosetTable.coset_table_max_limit.

Attributes of CosetTable

For CosetTable(G, H) where G is the group and H is the subgroup.

  • n: A non-negative integer, non-mutable attribute, dependently calculated as the maximum among the live-cosets (i.e. \(\Omega\)).
  • table: A list of lists, mutable attribute, mathematically represents the coset table.
  • omega: A list, dependent on the internal attribute p. \(\Omega\) represents the list of live-cosets. A standard coset-table has its \(\Omega = \{0, 1, \ldots, index-1 \}\) where \(index\) is the index of subgroup \(H\) in \(G\).

For experienced users we have a number of parameters that can be used to manipulate the algorithm, like

  • coset_table_max_limit (default value = \(4096000\)): manipulate the maximum number of cosets allowed in coset enumeration, i.e. the number of rows allowed in coset table. A coset enumeration will not finish if the subgroup does not have finite index, and even if it has it may take many more intermediate cosets than the actual index of the subgroup is. To avoid a coset enumeration “running away” therefore SymPy has a “safety stop” built-in. This is controlled by this variable. For example:

    >>> CosetTable.coset_table_max_limit = 50
    >>> F, a, b = free_group("a, b")
    >>> Cox = FpGroup(F, [a**6, b**6, (a*b)**2, (a**2*b**2)**2, (a**3*b**3)**5])
    >>> C_r = coset_enumeration_r(Cox, [a])
    Traceback (most recent call last):
      ...
    ValueError: the coset enumeration has defined more than 50 cosets
    
  • max_stack_size (default value = \(500\)): manipulate the maximum size of deduction_stack above or equal to which the stack is emptied.

Compression and Standardization

For any two entries \(i, j\) with \(i < j\) in coset table, the first occurrence of \(i\) in a coset table precedes the first occurrence of \(j\) with respect to the usual row-wise ordering of the table entries. We call such a table a standard coset table. To standardize a CosetTable we use the .standardize method.

Note the method alters the given table, it does not create a copy.

Subgroups of Finite Index

The functionality in this section are concerned with the construction of subgroups of finite index. We describe a method for computing all subgroups whose index does not exceed some (modest) integer bound.

Low Index Subgroups

low_index_subgroups(G, N): Given a finitely presented group \(G = \langle X \mid R \rangle\) (can be a free group), and N a positive integer, determine the conjugacy classes of subgroups of G whose indices is less than or equal to N.

For example to find all subgroups of \(G = \langle a, b \mid a^2 = b^3 = (ab)^4 = 1 \rangle\) having index <= 4, can be found as follows:

>>> from sympy.combinatorics.fp_groups import low_index_subgroups
>>> F, a, b = free_group("a, b")
>>> G = FpGroup(F, [a**2, b**3, (a*b)**4])
>>> l = low_index_subgroups(G, 4)
>>> for coset_table in l:
...     print(coset_table.table)
...
[[0, 0, 0, 0]]
[[0, 0, 1, 2], [1, 1, 2, 0], [3, 3, 0, 1], [2, 2, 3, 3]]
[[0, 0, 1, 2], [2, 2, 2, 0], [1, 1, 0, 1]]
[[1, 1, 0, 0], [0, 0, 1, 1]]

This returns the coset tables of subgroups of satisfying the property that index, \(index\), of subgroup in group is \(\le n\).

Constructing a presentation for a subgroup

In this section we discuss finding the presentation of a subgroup in a finitely presentation group. While the subgroup is currently allowed as input only in the form of a list of generators for the subgroup, you can expect the functionality of a coset table as input for subgroup in the group in near future.

There are two ways to construct a set of defining relations for subgroup from those of G. First is on a set of Schreier generators, known generally as Reidemeister-Schreier algorithm or on the given list of generators of H.

Reidemeister Schreier algorithm

called using reidemeister_presentation(G, Y) where G is the group and Y is a list of generators for subgroup H whose presentation we want to find.

>>> from sympy.combinatorics.fp_groups import reidemeister_presentation
>>> F, x, y = free_group("x, y")
>>> f = FpGroup(F, [x**3, y**5, (x*y)**2])
>>> H = [x*y, x**-1*y**-1*x*y*x]
>>> p1 = reidemeister_presentation(f, H)
>>> p1
((y_1, y_2), (y_1**2, y_2**3, y_2*y_1*y_2*y_1*y_2*y_1))

Bibliography

[CDHW73]John J. Cannon, Lucien A. Dimino, George Havas, and Jane M. Watson. Implementation and analysis of the Todd-Coxeter algorithm. Math. Comp., 27:463– 490, 1973.
[Ho05]Derek F. Holt, Handbook of Computational Group Theory. In the series ‘Discrete Mathematics and its Applications’, Chapman & Hall/CRC 2005, xvi + 514 p.
[Hav91]George Havas, Coset enumeration strategies. In Proceedings of the International Symposium on Symbolic and Algebraic Computation (ISSAC‘91), Bonn 1991, pages 191–199. ACM Press, 1991.