Elements

Elements

Elements in a finitely geenrated abelian group are of type GrpAbFinGenElem 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:

# gens — Method.

gens(G::GrpAbFinGen) -> Vector{GrpAbFinGenElem}

The sequence of generators of $G$.

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# GrpAbFinGen — Method.

(A::GrpAbFinGen)(x::Vector{fmpz}) -> GrpAbFinGenElem

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

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# GrpAbFinGen — Method.

(A::GrpAbFinGen)(x::fmpz_mat) -> GrpAbFinGenElem

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::GrpAbFinGen, i::Int) -> GrpAbFinGenElem

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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# rand — Method.

rand(G::GrpAbFinGen) -> GrpAbFinGenElem

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

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# rand — Method.

rand(G::GrpAbFinGen, B::fmpz) -> GrpAbFinGenElem

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::GrpAbFinGenElem) -> GrpAbFinGen

Returns the parent of $x$.

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Access

# getindex — Method.

getindex(x::GrpAbFinGenElem, i::Int) -> fmpz

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::GrpAbFinGenElem) -> fmpz

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(fmpz[1 2; 3 4])
(General) abelian group with relation matrix
[1 2; 3 4]

julia> for g = G
         println(g)
       end
Element of
(General) abelian group with relation matrix
[1 2; 3 4]
with structure of GrpAb: Z/2

with components [0 0]
Element of
(General) abelian group with relation matrix
[1 2; 3 4]
with structure of GrpAb: Z/2

with components [0 1]