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1 Introduction

This chapter serves as an introduction of the package LINS.

1.1 Overview

The package LINS provides an algorithm for computing the normal subgroups of a finitely presented group up to some given index bound.

Moreover it provides an interface for searching in the normal subgroup lattice of a finitely presented group. For example, one can use this interface to search for \(l\) normal subgroups of index \(n\).

The algorithm is based on work of David Firth [Fir05]. He implemented that algorithm in the algebra software MAGMA. That implementation in MAGMA has been revised and rewritten to a great extent by Derek Holt.

The current implementation in GAP uses a table of groups that was computed by the code in createTables.gi.

1.2 Examples

In this section we present example sessions which demonstrate how to use the main high-level functions provided by LINS.

1.2-1 Example : all normal subgroups up to index \(n\)

We compute all normal subgroups in \(D_{50}\), the dihedral group of size \(50\).

gap> G := DihedralGroup(50);
<pc group of size 50 with 3 generators>
gap> L := LowIndexNormalSubs(G, 50);;
gap> IsoTypes := List(L, StructureDescription);
[ "D50", "C25", "C5", "1" ]

1.2-2 Example : all normal subgroups of index \(n\)

We compute all normal subgroups of index \(5^2 = 25\) in \(C_5^4\), the direct product of \(4\) copies of the cyclic group of order \(5\):

gap> G := ElementaryAbelianGroup(5^4);
<pc group of size 625 with 4 generators>
gap> L := LowIndexNormalSubs(G, 5 ^ 2 : allSubgroups := false);;
gap> IsoTypes := Collected(List(L, StructureDescription));
[ [ "C5 x C5", 806 ] ]

1.3 Main Functions

In this section, we include all the main high-level functions provided to the user. For advanced search methods in the lattice of normal subgroups, take a look at Chapter 2.

1.3-1 LowIndexNormalSubs

Returns a list of all normal subgroups of G with index at most n. If the option allSubgroups is set to false, then only the normal subgroups of G with index equal to n are returned.

The generic method uses IsomorphismFpGroup (Reference: IsomorphismFpGroup) to transform G into an fp-group and then calls some variant of the low-level function LowIndexNormalSubgroupsSearch (2.3-1).

Note that a similar operation LowIndexNormalSubgroups (polycyclic: LowIndexNormalSubgroups) exists in the package polycyclic. Due to technical incompabilities, those operations could not be unified.

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