## 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 normliased 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:

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

The sequence of generators of $G$.

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

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

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randMethod
rand(G::FinGenAbGroup) -> FinGenAbGroupElem

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

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randMethod
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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parentMethod
parent(x::FinGenAbGroupElem) -> FinGenAbGroup

Returns the parent of $x$.

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### Access

getindexMethod
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

orderMethod
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 = G
println(g)
endAbelian group element [0, 0]
Abelian group element [0, 1]