Abelian Groups

Here we describe the interface to abelian groups in Hecke.

Introduction

Within Hecke, abelian groups are of generic abstract type GrpAb which does not have to be finitely generated, $\mathbb Q/\mathbb Z$ is an example of a more general abelian group. Having said that, most of the functionality is restricted to abelian groups that are finitely presented as $\mathbb Z$-modules.

Basic Creation

Finitely presented (as $\mathbb Z$-modules) abelian groups are of type GrpAbFinGen with elements of type GrpAbFinGenElem. The creation is mostly via a relation matrix $M = (m_{i,j})$ for $1\le i\le n$ and $1\le j\le m$. This creates a group with $m$ generators $e_j$ and relations

\[ \sum_{i=1}^n m_{i,j} e_j = 0.\]

abelian_group — Method

abelian_group(::Type{T} = GrpAbFinGen, M::ZZMatrix) -> GrpAbFinGen

Creates the abelian group with relation matrix M. That is, the group will have ncols(M) generators and each row of M describes one relation.

abelian_group — Method

abelian_group(::Type{T} = GrpAbFinGen, M::AbstractMatrix{<:IntegerUnion})

Creates the abelian group with relation matrix M. That is, the group will have ncols(M) generators and each row of M describes one relation.

abelian_group — Method

abelian_group(::Type{T} = GrpAbFinGen, M::AbstractMatrix{<:IntegerUnion})

Creates the abelian group with relation matrix M. That is, the group will have ncols(M) generators and each row of M describes one relation.

Alternatively, there are shortcuts to create products of cyclic groups:

abelian_group — Method

abelian_group(::Type{T} = GrpAbFinGen, M::AbstractVector{<:IntegerUnion}) -> GrpAbFinGen
abelian_group(::Type{T} = GrpAbFinGen, M::IntegerUnion...) -> GrpAbFinGen

Creates the direct product of the cyclic groups $\mathbf{Z}/m_i$, where $m_i$ is the $i$th entry of M.


julia> G = abelian_group(2, 2, 6)GrpAb: (Z/2)^2 x Z/6

or even

free_abelian_group — Method

free_abelian_group(::Type{T} = GrpAbFinGen, n::Int) -> GrpAbFinGen

Creates the free abelian group of rank n.

abelian_groups — Method

abelian_groups(n::Int) -> Vector{GrpAbFinGen}

Given a positive integer $n$, return a list of all abelian groups of order $n$.


julia> abelian_groups(8)3-element Vector{GrpAbFinGen}:
 GrpAb: (Z/2)^3
 GrpAb: Z/2 x Z/4
 GrpAb: Z/8

Invariants

is_snf — Method

is_snf(G::GrpAbFinGen) -> Bool

Return whether the current relation matrix of the group $G$ is in Smith normal form.

ngens — Method

ngens(G::GrpAbFinGen) -> Int

Return the number of generators of $G$ in the current representation.

nrels — Method

nrels(G::GrpAbFinGen) -> Int

Return the number of relations of $G$ in the current representation.

rels — Method

rels(A::GrpAbFinGen) -> ZZMatrix

Return the currently used relations of $G$ as a single matrix.

isfinite — Method

isfinite(A::GrpAbFinGen) -> Bool

Return whether $A$ is finite.

is_infinite — Method

is_infinite(x::Any) -> Bool

Tests whether $x$ is infinite, by returning !isfinite(x).

rank — Method

rank(A::GrpAbFinGen) -> Int

Return the rank of $A$, that is, the dimension of the $\mathbf{Q}$-vectorspace $A \otimes_{\mathbf Z} \mathbf Q$.

order — Method

order(A::GrpAbFinGen) -> ZZRingElem

Return the order of $A$. It is assumed that $A$ is finite.

exponent — Method

exponent(A::GrpAbFinGen) -> ZZRingElem

Return the exponent of $A$. It is assumed that $A$ is finite.

istrivial — Method

istrivial(A::GrpAbFinGen) -> Bool

Return whether $A$ is the trivial group.

is_torsion — Method

is_torsion(G::GrpAbFinGen) -> Bool

Return whether G is a torsion group.

is_cyclic — Method

is_cyclic(G::GrpAbFinGen) -> Bool

Return whether $G$ is cyclic.

elementary_divisors — Method

elementary_divisors(G::GrpAbFinGen) -> Vector{ZZRingElem}

Given $G$, return the elementary divisors of $G$, that is, the unique positive integers $e_1,\dotsc,e_k$ with $e_i \mid e_{i + 1}$ and $G \cong \mathbf{Z}/e_1\mathbf{Z} \times \dotsb \times \mathbf{Z}/e_k\mathbf{Z}$.