Elements

Elements in a finitely generated abelian group are of type FinGenAbGroupElem and are always given as a linear combination of the generators. Internally this representation is normalised to have a unique representative.

Creation

In addition to the standard function id, zero and one that can be used to create the neutral element, we also support more targeted creation:

gens Method

gens(G::FinGenAbGroup) -> Vector{FinGenAbGroupElem}

The sequence of generators of $G$.

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FinGenAbGroup Method

(A::FinGenAbGroup)(x::Vector{ZZRingElem}) -> FinGenAbGroupElem

Given an array x of elements of type ZZRingElem of the same length as ngens($A$), this function returns the element of $A$ with components x.

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FinGenAbGroup Method

(A::FinGenAbGroup)(x::ZZMatrix) -> FinGenAbGroupElem

Given a matrix over the integers with either $1$ row and ngens(A) columns or ngens(A) rows and $1$ column, this function returns the element of $A$ with components x.

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getindex Method

getindex(A::FinGenAbGroup, i::Int) -> FinGenAbGroupElem

Returns the element of $A$ with components $(0,\dots ,0,1,0,\dots ,0)$, where the $1$ is at the $i$-th position.

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rand Method

rand(G::FinGenAbGroup) -> FinGenAbGroupElem

Returns an element of $G$ chosen uniformly at random.

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rand Method

rand(G::FinGenAbGroup, B::ZZRingElem) -> FinGenAbGroupElem

For a (potentially infinite) abelian group $G$, return an element chosen uniformly at random with coefficients bounded by $B$.

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parent Method

parent(x::FinGenAbGroupElem) -> FinGenAbGroup

Returns the parent of $x$.

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Access

getindex Method

getindex(x::FinGenAbGroupElem, v::AbstractVector{Int}) -> Vector{ZZRingElem}

Returns the $i$-th components of the element $x$ where $i\in v$.

Note

This function is inefficient since the elements are internally stored using ZZMatrix but this function outputs a vector.

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getindex Method

getindex(x::FinGenAbGroupElem, i::Int) -> ZZRingElem

Returns the $i$-th component of the element $x$.

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Predicates

We have the standard predicates iszero, isone and is_identity to test an element for being trivial.

Invariants

order Method

order(A::FinGenAbGroupElem) -> ZZRingElem

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

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Iterator

One can iterate over the elements of a finite abelian group.

julia> G = abelian_group(ZZRingElem[1 2; 3 4])
Finitely generated abelian group
  with 2 generators and 2 relations and relation matrix
  [1   2]
  [3   4]

julia> for g in G
         println(g)
       end
Abelian group element [0, 0]
Abelian group element [0, 1]