Introduction

Functions to work with finitely presented nilpotent groups. To install al the functions save the files inside the GAP directory/pkg/NilGroups and then run in GAP.

Read( "*Gap Directory*/pkg/NilGroups/read.g");

Some functions in this repository rely on the NQ package, so having it installed is mandatory.

Contents

read.g: File to read in gap.

nilgrp.gd: Definition of the functions.

general.gi: Offers some functions that are general for the other files.

exam.gi: Functions to generate the groups used in experiments.

• SomeNilpotentGroups( n ) : Retruns the group assigned to the number n, this groups are:
  • n = 1 : Group Torsion-free of Hirsch length 5.
  • n = 2 : Group with relative orders [0, 0, 0, 0, 0, 4, 2, 2].
  • n = 3 : Group with relative orders [0, 0, 0, 0, 3872].
  • n = 4 : Group with relative orders [0, 0, 0, 352, 11264].
  • n = 5 : Group with 85 generators and Hirsch length 23.
  • n = 6 : Group with 72 generators and Hirsch length 37.
  • n = 7 : Group with 20 generators and Hirsch length 6.
  • n = 8 : Group with 21 generators and Hirsch length 10.

elmcon.gi: Functions to solve the element conjugacy problems:

• Centralizer problem
  • CentralizerNilGroup( G, elms ) : Returns the centralizer of the elements in the list elms.
• Conjugacy problem
  • IsConjugateNilGroup( G, g, h ) : If the elements g and h are conjugate returns the conjugating element, otherwise returns false.
  • IsCanonicalConjugateElements( G, elms ) : If the elements of the list elms are conjugate returns the canonical element, kano, the conjugating element, conj, and the centralizer cent. Otherwise returns false.
• Canonical conjugate
  • CanonicalConjugateElements( G, elms ) : Returns the canonical conjugate of each element in the list elms. Also returs the centralizers and the conjugating elements.
• Conjugacy problem for lists of elements
  • IsConjugateList( G, list ) : Given a list of elements in G, returns a conjugate representative for each conjugacy class found and a list of indexes which represents the elements that belong to the respective conjugacy class.
  • CanonicalConjugateList( G, list ) : Given a list of elements in G, returns a conjugate representative for each conjugacy class found and a list of indexes which represents the elements that belong to the respective conjugacy class.

Setting the information level to 1 of InfoConjugacy will show the progres of the algoritms.

subgrpcon.gi: Functions to solve the subgroup conjugacy problems:

• Normalizer problem
  • NormalizerNilGroup( G, U ) : Returns the normalizer of the subgroup U in the group G.
• Subgroup Conjugacy problem
  • IsConjugateSubgroups( G, U, V ) : Returns the conjugating element if the two subgroups are conjugated. Otherwise returns false.
  • IsCanonicalConjugateSubgroups( G, U, V ) : Returns the canonical conjugate of U and V, the conjugating elements and the normalizer, if both subgroups are conjugated. Otherwise reutrns false.
• Canonical conjugate subgroup
  • CanonicalConjugateSubgroup( G, U ) : Returns the canonical conjugate of U, the conjugating element and the normalizer.

Setting the information level to 1 of InfoConjugacy will show the progres of the algoritms.

inter.gi: Intersection functions.

• IntersectionSubgroupsNilGroups( G, U, V ) : Returns a list of generators for the intersection of U and V.

prod.gi: Product functions.

• SubgroupProductPair( G, U, V ) : Returns a product pair of U and V.
• ProductDecomposition( G, U, V, g) : Returns, if posible, a descomposition of g as a multiplication of elements in U and V

experiments.g:

Script to see the performance of the algorithms using the groups in exam.gi. The results are avaribale in [B. Eick and O. F. Ayala].

Cites

• Eddie H. Lo. Finding intersections and normalizers in finitely generated nilpotent groups. J. Symbolic Comput., 25(1):45–59, 1998.
• Charles C. Sims. Computation with finitely presented groups, volume 48 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1994.
• B. Eick and G. Ostheimer. On the orbit-stabilizer problem for integral matrix actions of polycyclic groups. Mathematics of Computation,72(243):1511–1529, 2003.
• B. Eick and O. F. Ayala. The conjugacy problem and canonical representatives in finitely generated nilpotent groups.