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::fmpz_mat) -> 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.

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# 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.

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# 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.

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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.

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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.

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# abelian_groups — Method.

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

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{GrpAbFinGen}:
 GrpAb: (Z/2)^3
 GrpAb: Z/2 x Z/4
 GrpAb: Z/8

Invariants

# is_snf — Method.

is_snf(G::GrpAbFinGen) -> Bool

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

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# ngens — Method.

ngens(G::GrpAbFinGen) -> Int

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

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# nrels — Method.

nrels(G::GrpAbFinGen) -> Int

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

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# rels — Method.

rels(A::GrpAbFinGen) -> fmpz_mat

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

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# isfinite — Method.

isfinite(A::GrpAbFinGen) -> Bool

Returns whether $A$ is finite.

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# is_infinite — Method.

is_infinite(x::Any) -> Bool

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

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# rank — Method.

rank(A::GrpAbFinGen) -> Int

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

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# order — Method.

order(A::GrpAbFinGen) -> fmpz

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

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# exponent — Method.

exponent(A::GrpAbFinGen) -> fmpz

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

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# istrivial — Method.

istrivial(A::GrpAbFinGen) -> Bool

Checks if $A$ is the trivial group.

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# is_torsion — Method.

is_torsion(G::GrpAbFinGen) -> Bool

Returns true if and only if G is a torsion group.

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# is_cyclic — Method.

is_cyclic(G::GrpAbFinGen) -> Bool

Returns whether $G$ is cyclic.

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# elementary_divisors — Method.

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

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

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