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 $e_1,\dots ,e_k$ with $e_i\mid e_{i+1}$ and $e_i\ne 1$ such that $G\cong \mathbf{Z}/e_1\mathbf{Z}\times \cdots \times \mathbf{Z}/e_k\mathbf{Z}$.

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