2 Introduction

Sections

1. Overview
2. Background on (polycyclic) parametrised presentations
3. Computation of Schur multiplicators
4. Computation of low-dimensional cohomology
5. Example

2.1 Overview

The coclass of a finite p-group of order pⁿ and nilpotency class c is defined as n−c. This invariant of finite p-groups has been introduced by Leedham-Green and Newman in LGN80 and it became of major importance in p-group theory.

A first tool in the classification of all p-groups of coclass r is the coclass graph G(p,r). Its vertices are the isomorphism types of finite p-groups of coclass r. Two vertices G and H are joined by an edge if G is isomorphic to the quotient H/γ(H) where γ(H) is the last non-trivial term of the lower series of H.

Du Sautoy dS00 and Eick and Leedham-Green ELG08 proved that G(p,r) contains certain periodic patterns. Eick and Leedham-Green ELG08 define infinite coclass sequences of finite p-groups of coclass r which underpin this periodic pattern. In G(2,r) and G(3,1) almost all groups are contained in an infinite coclass sequence.

Eick and Leedham-Green ELG08 also proved that the infinitely many p-groups in an infinite coclass sequence can be defined by a single parametrised presentation.

The first aim of this package is the definition of polycyclic parametrised presentations; these are parametrised presentations as defined by Eick and Leedham-Green ELG08 and additionally they have various features of polycyclic presentations. Each such presentation defines all the infinitely many finite p-groups in an infinite coclass sequence.

We then provide some algorithms to compute with polycyclic parametrised presentations. In particular, we introduce a generalisation of the collection algorithm for polycyclic parametrised presentations. Based on this, we describe algorithms to compute polycyclic parametrised presentations for Schur extensions, for the Schur multiplicator and for some low-dimensional cohomology groups. We refer to EF11 for details on the underlying algorithms and further references.

Finally, we exhibit a database of polycyclic parametrised presentations for the infinite coclass families of the finite 2-groups of coclass at most 2 and the finite 3-groups of coclass 1.

2.2 Background on (polycyclic) parametrised presentations

In this section we describe the polycyclic parametrised presentations (pp-presentations) for infinite coclass sequences.

Let (G_x | x ∈ N), where N denotes the natural numbers, be an infinite coclass sequence; x is the parameter of this infinite coclass sequence. Then every group G_x is an extension of a finite p-group P of order pⁿ by an abelian p-group T_x of rank d. Furthermore, every G_x has a polycyclic presentation (short pp-presentation) on generators g₁, …, g_n, t₁, …, t_d with relations of the form

gip = gi+1ai,i,i+1 …gnai,i,nt1αi,i,1(x) …tdαi,i,d(x), gigj = gj+1ai,j,j+1 …gnai,j,nt1αi,j,1(x) …tdαi,j,d(x), tkgi = t1bk,i,1(x) …tdbk,i,d(x), tktl = tk, tkpx+e = 1,

where 1 ≤ j < i ≤ n and 1 ≤ k < l ≤ d; certain a_i,j,m ∈ {0, …, p−1}, a non-negative integer e, α_k,l,m(x) of the form c_k,l,m+p^xd_k,l,m and b_k,l,m with b_k,l,m,c_k,l,m,d_k,l,m certain p-adic integers. The p-adic exponents arising in the relations can be reduced modulo the relative orders of the involved elements and thus can be reduced to integers for every specific x.

We call such a pp-presentation integral if all the p-adic numbers b_k,l,m, c_k,l,m, d_k,l,m are integers. Our algorithms introduced in this package compute with integral pp-presentations only.

We call such an pp-presentation consistent if for every x ∈ N the presentation is consistent as a polycyclic presentation; where we possibly reduce the exponents in the presentation modulo the relative orders of the generators.

2.3 Computation of Schur multiplicators

In this section we recall briefly the method of EF11 to determine the Schur multiplicators of almost all groups G_x in an infinite coclass sequence.

Suppose we are given a consistent integral pp-presentation F/R_x for the groups G_x in an infinite coclass sequence, where F is a free group and R_x is generated by parametrised relations as above. Note that the exponents in these relations depend on x, while the number of generators and the number of relations does not depend on the parameter.

Using this presentation we can define a parametrised presentation for the Schur extensions G_x^* = F/[F,R_x], corresponding to the parametrised presentation F/R_x. The next step is to find the isomorphism types of Y_x = R_x/[F,R_x] since M(G_x) ≅ (F′∩R_x)/[F,R_x] are the torsion subgroups of Y_x as all G_x are finite p-groups.

Then Y_x = R_x/[F,R_x] are generated by certain so-called consistency relations. Using this we can compute the isomorphism types of Y_x and thus the isomorphism types of M(G_x) for almost all G_x in the chosen infinite coclass sequence.

2.4 Computation of low-dimensional cohomology

From the parametrised presentation F/R_x we can see that the Abelian invariants are the same for all groups G_x in an infinite coclass sequence, and we can compute them. Using this and the computation of the Schur multiplicators one obtains Hⁿ(G_x,Z) and Hⁿ(G_x,GF(p)) for 0 ≤ n ≤ 2, where the G_x act trivially on Z and GF(p), respectively.

2.5 Example

In this section we present the well-known example of quaternion groups Q_2^x+3. It is well known that they have a parametrised presentation of the following form:

{ g1,g2,t1 |  g12 = t12x,  g2g1 = g2t1−1+2x+1,  g22 = t1,  t1g1 = t1−1+2x+1,  t12x+1 = 1 }·

Using this we can define the Schur extensions Q2x+3*

{ g1,g2,t1,c1, c2, c3 | g12 = t12xc3,  g2g1 = g2t1−1+2x+1c21−2x+1,  g22 = t1c1, t1g1 = t1−1+2x+1c22−2x+1,  t12x+1 = c22x+1,  c1,c2,c3 central }·

This yields M(Q_2^x+3) = 1.

[Up] [Previous] [Next] [Index]