Contents

Polycyclic Groups

Polycyclic Groups

Introduction

This module presents the functionality designed for computing with polycyclic groups(PcGroup for short). The name of the corresponding SymPy object is PolycyclicGroup. The functions or classes described here are studied under Computational Group Theory.

Overview of functionalities

  • The construction of PolycyclicGroup from a given PermutationGroup.

  • Computation of polycyclic generating sequence(pcgs for short) and polycyclic series(pc_series).

  • Computation of relative order for polycyclic series.

  • Implementation of class Collector which can be treated as a base for polycylic groups.

  • Implementation of polycyclic group presentation(pc_presentation for short).

  • Computation of exponent vector, depth and leading exponent for a given element of a polycyclic group.

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

The Construction of Polycyclic Groups

Given a Permutation Group, A Polycyclic Group is constructed by computing the corresponding polycylic generating sequence, polycyclic series and it’s relative order.

Attributes of PolycyclicGroup

  • pc_sequence : Polycyclic sequence is formed by collecting all the missing generators between the adjacent groups in the derived series of given permutation group.

  • pc_series : Polycyclic series is formed by adding all the missing generators of der[i+1] in der[i], where der represents derived series.

  • relative_order : A list, computed by the ratio of adjacent groups in pc_series.

  • collector : By default, it is None. Collector class provides the polycyclic presentation.

>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> G = SymmetricGroup(4)
>>> PcGroup = G.polycyclic_group()
>>> len(PcGroup.pcgs)
4
>>> pc_series = PcGroup.pc_series
>>> pc_series[0].equals(G)  # use equals, not literal `==`
True
>>> gen = pc_series[len(pc_series) - 1].generators[0]
>>> gen.is_identity
True
>>> PcGroup.relative_order
[2, 3, 2, 2]

The Construction of Collector

Collector is one of the attributes of class PolycyclicGroup.

Attributes of Collector

Collector posses all the attributes of PolycyclicGroup, In addition there are few more attributes which are defined below:

  • free_group : free_group provides the mapping of polycyclic generating sequence with the free group elements.

  • pc_presentation : Provides the presentation of polycyclic groups with the help of power and conjugate relators.

>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> G = SymmetricGroup(3)
>>> PcGroup = G.polycyclic_group()
>>> Collector = PcGroup.collector
>>> Collector.free_group
<free group on the generators (x0, x1)>
>>> Collector.pc_presentation
{x0**2: (), x1**3: (), x0**-1*x1*x0: x1**2}

Computation of Minimal Uncollected Subword

A word V defined on generators in the free_group of pc_group is a minimal uncollected subword of the word W if V is a subword of W and it has one of the following form:

  • \(v = {x_{i+1}}^{a_j}x_i\)

  • \(v = {x_{i+1}}^{a_j}{x_i}^{-1}\)

  • \(v = {x_i}^{a_j}\)

\(a_j \notin \{0, \ldots \mathrm{relative\_order}[j]-1\}\).

>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> from sympy.combinatorics.free_groups import free_group
>>> G = SymmetricGroup(4)
>>> PcGroup = G.polycyclic_group()
>>> collector = PcGroup.collector
>>> F, x1, x2 = free_group("x1, x2")
>>> word = x2**2*x1**7
>>> collector.minimal_uncollected_subword(word)
((x2, 2),)

Computation of Subword Index

For a given word and it’s subword, subword_index computes the starting and ending index of the subword in the word.

>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> from sympy.combinatorics.free_groups import free_group
>>> G = SymmetricGroup(4)
>>> PcGroup = G.polycyclic_group()
>>> collector = PcGroup.collector
>>> F, x1, x2 = free_group("x1, x2")
>>> word = x2**2*x1**7
>>> w = x2**2*x1
>>> collector.subword_index(word, w)
(0, 3)
>>> w = x1**7
>>> collector.subword_index(word, w)
(2, 9)

Computation of Collected Word

A word W is called collected, if W \(= {x_{i_1}}^{a_1} \ldots {x_{i_r}}^{a_r}\) with \(i_1 < i_2< \ldots < i_r\) and \(a_j\) is in \(\{1 \ldots s_{j-1}\}\), where \(s_j\) represents the respective relative order.

>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> from sympy.combinatorics.free_groups import free_group
>>> G = SymmetricGroup(4)
>>> PcGroup = G.polycyclic_group()
>>> collector = PcGroup.collector
>>> F, x0, x1, x2, x3 = free_group("x0, x1, x2, x3")
>>> word = x3*x2*x1*x0
>>> collected_word = collector.collected_word(word)
>>> free_to_perm = {}
>>> free_group = collector.free_group
>>> for sym, gen in zip(free_group.symbols, collector.pcgs):
...     free_to_perm[sym] = gen
>>> G1 = PermutationGroup()
>>> for w in word:
...     sym = w[0]
...     perm = free_to_perm[sym]
...     G1 = PermutationGroup([perm] + G1.generators)
>>> G2 = PermutationGroup()
>>> for w in collected_word:
...     sym = w[0]
...     perm = free_to_perm[sym]
...     G2 = PermutationGroup([perm] + G2.generators)

The two are not identical but they are equivalent:

>>> G1 == G2
False
>>> G1.equals(G2)
True

Computation of Polycyclic Presentation

The computation of presentation starts from the bottom of the pcgs and polycyclic series. Storing all the previous generators from pcgs and then taking the last generator as the generator which acts as a conjugator and conjugates all the previous generators in the list.

To get a clear picture, start with an example of SymmetricGroup(4). For S(4) there are 4 generators in pcgs say \([x_0, x_1, x_2, x_3]\) and the relative_order vector is [2, 3, 2, 2]. Starting from bottom of this sequence the presentation is computed in order as below.

using only \([x_3]\) from pcgs and pc_series[4] compute:

  • \(x_3^2\)

using only \([x_3]\) from pcgs and pc_series[3] compute:

  • \(x_2^2\)

  • \(x_2^{-1}x_3x_2\)

using \([x_3, x_2]\) from pcgs and pc_series[2] compute:

  • \(x_1^3\)

  • \(x_1^{-1}x_3x_1\)

  • \(x_1^{-1}x_2x_1\)

using \([x_3, x_2, x_1]\) from pcgs and pc_series[1] compute:

  • \(x_0^2\)

  • \(x_0^{-1}x_3x_0\)

  • \(x_0^{-1}x_2x_0\)

  • \(x_0^{-1}x_1x_0\)

One thing to note is same group can have different pcgs due to variying derived_series which, results in different polycyclic presentations.

>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> from sympy.combinatorics.permutations import Permutation
>>> G = SymmetricGroup(4)
>>> PcGroup = G.polycyclic_group()
>>> collector = PcGroup.collector
>>> pcgs = PcGroup.pcgs
>>> len(pcgs)
4
>>> free_group = collector.free_group
>>> pc_resentation = collector.pc_presentation
>>> free_to_perm = {}
>>> for s, g in zip(free_group.symbols, pcgs):
...     free_to_perm[s] = g
>>> for k, v in pc_resentation.items():
...     k_array = k.array_form
...     if v != ():
...        v_array = v.array_form
...     lhs = Permutation()
...     for gen in k_array:
...         s = gen[0]
...         e = gen[1]
...         lhs = lhs*free_to_perm[s]**e
...     if v == ():
...         assert lhs.is_identity
...         continue
...     rhs = Permutation()
...     for gen in v_array:
...         s = gen[0]
...         e = gen[1]
...         rhs = rhs*free_to_perm[s]**e
...     assert lhs == rhs

Computation of Exponent Vector

Any generator of the polycyclic group can be represented with the help of it’s polycyclic generating sequence. Hence, the length of exponent vector is equal to the length of the pcgs.

A given generator g of the polycyclic group, can be represented as \(g = x_1^{e_1} \ldots x_n^{e_n}\), where \(x_i\) represents polycyclic generators and n is the number of generators in the free_group equal to the length of pcgs.

>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> from sympy.combinatorics.permutations import Permutation
>>> G = SymmetricGroup(4)
>>> PcGroup = G.polycyclic_group()
>>> collector = PcGroup.collector
>>> pcgs = PcGroup.pcgs
>>> collector.exponent_vector(G[0])
[1, 0, 0, 0]
>>> exp = collector.exponent_vector(G[1])
>>> g = Permutation()
>>> for i in range(len(exp)):
...     g = g*pcgs[i]**exp[i] if exp[i] else g
>>> assert g == G[1]

Depth of Polycyclic generator

Depth of a given polycyclic generator is defined as the index of the first non-zero entry in the exponent vector.

>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> G = SymmetricGroup(3)
>>> PcGroup = G.polycyclic_group()
>>> collector = PcGroup.collector
>>> collector.depth(G[0])
2
>>> collector.depth(G[1])
1

Computation of Leading Exponent

Leading exponent represents the exponent of polycyclic generator at the above depth.

>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> G = SymmetricGroup(3)
>>> PcGroup = G.polycyclic_group()
>>> collector = PcGroup.collector
>>> collector.leading_exponent(G[1])
1

Bibliography

Ho05

Derek F. Holt, Handbook of Computational Group Theory. In the series ‘Discrete Mathematics and its Applications’, Chapman & Hall/CRC 2005, xvi + 514 p.

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