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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_group — Method
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_group — Method
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_group — Method
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_group — Method
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_group — Method
free_abelian_group(::Type{T} = FinGenAbGroup, n::Int) -> FinGenAbGroup

Creates the free abelian group of rank n.

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abelian_groups — Method
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_snf — Method
is_snf(G::FinGenAbGroup) -> Bool

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

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number_of_generators — Method
number_of_generators(G::FinGenAbGroup) -> Int

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

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nrels — Method
number_of_relations(G::FinGenAbGroup) -> Int

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

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rels — Method
rels(A::FinGenAbGroup) -> ZZMatrix

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

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is_finite — Method
isfinite(A::FinGenAbGroup) -> Bool

Return whether $A$ is finite.

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torsion_free_rank — Method
torsion_free_rank(A::FinGenAbGroup) -> Int

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

See also rank.

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order — Method
order(A::FinGenAbGroup) -> ZZRingElem

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

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exponent — Method
exponent(A::FinGenAbGroup) -> ZZRingElem

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

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is_trivial — Method
is_trivial(A::FinGenAbGroup) -> Bool

Return whether $A$ is the trivial group.

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is_torsion — Method
is_torsion(G::FinGenAbGroup) -> Bool

Return whether G is a torsion group.

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is_cyclic — Method
is_cyclic(G::FinGenAbGroup) -> Bool

Return whether $G$ is cyclic.

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elementary_divisors — Method
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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