# 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 FinGenAbGroup with elements of type FinGenAbGroupElem. 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_groupMethod
abelian_group(::Type{T} = FinGenAbGroup, M::ZZMatrix) -> FinGenAbGroup

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.

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abelian_groupMethod
abelian_group(::Type{T} = FinGenAbGroup, 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.

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abelian_groupMethod
abelian_group(::Type{T} = FinGenAbGroup, 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.

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

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

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

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

or even

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

Creates the free abelian group of rank n.

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abelian_groupsMethod
abelian_groups(n::Int) -> Vector{FinGenAbGroup}

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

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

### Invariants

is_snfMethod
is_snf(G::FinGenAbGroup) -> Bool

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

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nrelsMethod
number_of_relations(G::FinGenAbGroup) -> Int

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

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relsMethod
rels(A::FinGenAbGroup) -> ZZMatrix

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

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orderMethod
order(A::FinGenAbGroup) -> ZZRingElem

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

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exponentMethod
exponent(A::FinGenAbGroup) -> ZZRingElem

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

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is_torsionMethod
is_torsion(G::FinGenAbGroup) -> Bool

Return whether G is a torsion group.

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elementary_divisorsMethod
elementary_divisors(G::FinGenAbGroup) -> Vector{ZZRingElem}

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

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