Polycyclic groups

Introduction

A polycyclic group is a group $G$ which has a subnormal series $G = G_1 > G_2 > \cdots > G_n > G_{n+1} = \{ 1 \}$ such that the factor groups $G_i/G_{i+1}$ are cyclic.

Polycyclic groups are solvable, and every finite solvable group is polycyclic.

Each polycyclic group has a finite presentation with the following structure: Choose elements $g_i \in G_i \setminus G_{i+1}$, for $1 \leq i \leq n$, such that $G_i = \langle G_{i+1}, g_i \rangle$ holds, and let $r_i$ be the index of $G_{i+1}$ in $G_i$ (a positive integer or infinity), the relative order of $g_i$ w.r.t. the given subnormal series. Then each element of $G$ has a unique expression of the form $g_1^{e_1} g_2^{e_2} \cdots g_n^{e_n}$ where the exponent $e_i$ is in the set $\{ 0, 1, \ldots, r_i-1 \}$ if $r_i$ is finite, and $e_i$ is an integer otherwise; this expression is called the normal form of the group element.

The group $G$ has a presentation with generators $g_1, g_2, \ldots, g_n$ and defining relations of the following form, where we define $I$ as the set of indices $i$ for which $r_i$ is a positive integer.

\[ \begin{array}{lcll} g_i^{r_i} & = & g_{i+1}^{a(i,i,i+1)} \cdots g_n^{a(i,i,n)}, & 1 \leq i \leq n, i \in I \\ g_i^{-1} g_j g_i & = & g_{i+1}^{a(i,j,i+1)} \cdots g_n^{a(i,j,n)}, & 1 \leq i < j \leq n. \end{array}\]

For infinite groups, we need additionally

\[ \begin{array}{lcll} g_i^{-1} g_j^{-1} g_i & = & g_{i+1}^{b(i,j,i+1)} \cdots g_n^{b(i,j,n)}, & 1 \leq i < j \leq n, j \not\in I \\ g_i g_j g_i^{-1} & = & g_{i+1}^{c(i,j,i+1)} \cdots g_n^{c(i,j,n)}, & 1 \leq i < j \leq n, i \not\in I \\ g_i g_j^{-1} g_i^{-1} & = & g_{i+1}^{d(i,j,i+1)} \cdots g_n^{d(i,j,n)}, & 1 \leq i < j \leq n, i, j, \not\in I. \end{array}\]

Here we assume that the right hand sides are words in normal form.

This presentation describes a confluent rewriting system, thus it admits the computation of the normal form for each group element. Inverses and products of elements given by their normal forms can be efficiently written in normal form.

Groups which are given by a polycyclic presentation are called polycyclicly presented groups. The rest of this section is about these groups.

Polycyclicly presented groups in Oscar have the type PcGroup, their elements have the type PcGroupElem. Analogous to the situation with finitely presented groups and their subgroups, there are the types SubPcGroup for subgroups and SubPcGroupElem for their elements.

Basic Creation

One can write down a polycyclic presentation by hand, using a so-called collector, and then creating a group with this presentation.

julia> c = collector(2, Int);

julia> set_relative_orders!(c, [2, 3])

julia> set_conjugate!(c, 2, 1, [2 => 2])

julia> gg = pc_group(c)
Pc group of order 6

julia> describe(gg)
"S3"

Alternatively, one can take a polycyclic group, and let Oscar compute a pc presentation for it.

julia> g = symmetric_group(4)
Sym(4)

julia> iso = isomorphism(PcGroup, g);

julia> h = codomain(iso)
Pc group of order 24

For certain series of groups, one can create the members as pc groups.

julia> dihedral_group(8)
Pc group of order 8

And the groups from the small groups library are represented by pc groups whenever they are solvable.

julia> small_group(24, 12)
Pc group of order 24

Functions for (subgroups of) pc groups and their elements

relators — Method

relators(G::PcGroup)

Return a vector of elements in a free group of rank ngens(G) that describes the defining relators of the underlying polycyclic presentation of G.

Examples

julia> g = dihedral_group(8)
Pc group of order 8

julia> relators(g)
6-element Vector{FPGroupElem}:
 g1^2
 g2^-1*g1^-1*g2*g1*g3^-1
 g3^-1*g1^-1*g3*g1
 g2^2*g3^-1
 g3^-1*g2^-1*g3*g2
 g3^2

Polycyclic groups

Introduction

Basic Creation

Functions for (subgroups of) pc groups and their elements

Series of polycyclic groups

Collectors for polycyclicly presented groups

Technicalities