326 lines
9.9 KiB
Julia
326 lines
9.9 KiB
Julia
import Base: ×
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export DirectProductGroup, DirectProductGroupElem
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export MultiplicativeGroup, MltGrp, MltGrpElem
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export AdditiveGroup, AddGrp, AddGrpElem
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###############################################################################
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#
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# MltGrp/MltGrpElem & AddGrp/AddGrpElem
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# (a thin wrapper for multiplicative/additive group of a Ring)
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#
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###############################################################################
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for (Gr, Elem) in [(:MltGrp, :MltGrpElem), (:AddGrp, :AddGrpElem)]
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@eval begin
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struct $Gr{T<:AbstractAlgebra.Ring} <: AbstractAlgebra.Group
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obj::T
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end
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struct $Elem{T<:AbstractAlgebra.RingElem} <: AbstractAlgebra.GroupElem
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elt::T
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end
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==(g::$Elem, h::$Elem) = g.elt == h.elt
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==(G::$Gr, H::$Gr) = G.obj == H.obj
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elem_type(::Type{$Gr{T}}) where T = $Elem{elem_type(T)}
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parent_type(::Type{$Elem{T}}) where T = $Gr{parent_type(T)}
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parent(g::$Elem) = $Gr(parent(g.elt))
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end
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end
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MultiplicativeGroup = MltGrp
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AdditiveGroup = AddGrp
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(G::MltGrp)(g::MltGrpElem) = MltGrpElem(G.obj(g.elt))
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function (G::MltGrp)(g)
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r = (G.obj)(g)
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isunit(r) || throw(DomainError(
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"Cannot coerce to multplicative group: $r is not invertible!"))
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return MltGrpElem(r)
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end
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(G::AddGrp)(g) = AddGrpElem((G.obj)(g))
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(G::MltGrp)() = MltGrpElem(G.obj(1))
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(G::AddGrp)() = AddGrpElem(G.obj())
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inv(g::MltGrpElem) = MltGrpElem(inv(g.elt))
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inv(g::AddGrpElem) = AddGrpElem(-g.elt)
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for (Elem, op) in ([:MltGrpElem, :*], [:AddGrpElem, :+])
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@eval begin
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^(g::$Elem, n::Integer) = $Elem(op(g.elt, n))
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function *(g::$Elem, h::$Elem)
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parent(g) == parent(h) || throw(DomainError(
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"Cannot multiply elements of different parents"))
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return $Elem($op(g.elt,h.elt))
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end
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end
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end
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Base.show(io::IO, G::MltGrp) = print(io, "The multiplicative group of $(G.obj)")
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Base.show(io::IO, G::AddGrp) = print(io, "The additive group of $(G.obj)")
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Base.show(io::IO, g::Union{MltGrpElem, AddGrpElem}) = show(io, g.elt)
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gens(F::AbstractAlgebra.Field) = elem_type(F)[gen(F)]
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order(G::AddGrp{<:AbstractAlgebra.GFField}) = order(G.obj)
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elements(G::AddGrp{F}) where F <: AbstractAlgebra.GFField = (G((i-1)*G.obj(1)) for i in 1:order(G))
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order(G::MltGrp{<:AbstractAlgebra.GFField}) = order(G.obj) - 1
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elements(G::MltGrp{F}) where F <: AbstractAlgebra.GFField = (G(i*G.obj(1)) for i in 1:order(G))
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###############################################################################
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#
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# DirectProductGroup / DirectProductGroupElem
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#
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###############################################################################
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doc"""
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DirectProductGroup(G::Group, n::Int) <: Group
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Implements `n`-fold direct product of `G`. The group operation is
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`*` distributed component-wise, with component-wise identity as neutral element.
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"""
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struct DirectProductGroup{T<:Group} <: Group
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group::T
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n::Int
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end
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struct DirectProductGroupElem{T<:GroupElem} <: GroupElem
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elts::Vector{T}
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end
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###############################################################################
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#
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# Type and parent object methods
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#
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###############################################################################
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elem_type(::Type{DirectProductGroup{T}}) where {T} =
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DirectProductGroupElem{elem_type(T)}
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parent_type(::Type{DirectProductGroupElem{T}}) where {T} =
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DirectProductGroup{parent_type(T)}
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parent(g::DirectProductGroupElem) =
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DirectProductGroup(parent(first(g.elts)), length(g.elts))
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###############################################################################
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#
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# AbstractVector interface
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#
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###############################################################################
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Base.size(g::DirectProductGroupElem) = size(g.elts)
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Base.IndexStyle(::Type{DirectProductGroupElem}) = Base.LinearFast()
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Base.getindex(g::DirectProductGroupElem, i::Int) = g.elts[i]
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function Base.setindex!(g::DirectProductGroupElem{T}, v::T, i::Int) where {T}
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parent(v) == parent(g.elts[i]) || throw(DomainError(
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"$g is not an element of $i-th factor of $(parent(G))"))
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g.elts[i] = v
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return g
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end
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function Base.setindex!(g::DirectProductGroupElem{T}, v::S, i::Int) where {T, S}
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g.elts[i] = parent(g.elts[i])(v)
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return g
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end
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###############################################################################
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#
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# DirectProductGroup / DirectProductGroupElem constructors
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#
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###############################################################################
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function ×(G::Group, H::Group)
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G == H || throw(DomainError(
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"Direct Powers are defined only for the same groups"))
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return DirectProductGroup(G,2)
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end
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×(H::Group, G::DirectProductGroup) = G×H
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function ×(G::DirectProductGroup, H::Group)
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G.group == H || throw(DomainError(
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"Direct products are defined only for the same groups"))
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return DirectProductGroup(G.group,G.n+1)
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end
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DirectProductGroup(R::T, n::Int) where {T<:AbstractAlgebra.Ring} =
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DirectProductGroup(AdditiveGroup(R), n)
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function ×(G::DirectProductGroup{T}, H::Group) where T <: Union{AdditiveGroup, MultiplicativeGroup}
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G.group == T(H) || throw(DomainError(
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"Direct products are defined only for the same groups"))
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return DirectProductGroup(G.group,G.n+1)
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end
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###############################################################################
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#
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# Parent object call overloads
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#
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###############################################################################
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doc"""
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(G::DirectProductGroup)(a::Vector, check::Bool=true)
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> Constructs element of the $n$-fold direct product group `G` by coercing each
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> element of vector `a` to `G.group`. If `check` flag is set to `false` neither
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> check on the correctness nor coercion is performed.
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"""
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function (G::DirectProductGroup)(a::Vector, check::Bool=true)
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if check
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G.n == length(a) || throw(DomainError(
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"Can not coerce to DirectProductGroup: lengths differ"))
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a = (G.group).(a)
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end
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return DirectProductGroupElem(a)
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end
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(G::DirectProductGroup)() = DirectProductGroupElem([G.group() for _ in 1:G.n])
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(G::DirectProductGroup)(g::DirectProductGroupElem) = G(g.elts)
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(G::DirectProductGroup)(a::Vararg{T, N}) where {T, N} = G([a...])
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###############################################################################
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#
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# Basic manipulation
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#
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###############################################################################
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function hash(G::DirectProductGroup, h::UInt)
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return hash(G.group, hash(G.n, hash(DirectProductGroup,h)))
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end
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function hash(g::DirectProductGroupElem, h::UInt)
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return hash(g.elts, hash(parent(g), hash(DirectProductGroupElem, h)))
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end
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###############################################################################
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#
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# String I/O
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#
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###############################################################################
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function show(io::IO, G::DirectProductGroup)
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print(io, "$(G.n)-fold direct product of $(G.group)")
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end
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function show(io::IO, g::DirectProductGroupElem)
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print(io, "[$(join(g.elts,","))]")
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end
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###############################################################################
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#
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# Comparison
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#
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###############################################################################
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doc"""
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==(g::DirectProductGroup, h::DirectProductGroup)
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> Checks if two direct product groups are the same.
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"""
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function (==)(G::DirectProductGroup, H::DirectProductGroup)
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G.group == H.group || return false
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G.n == G.n || return false
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return true
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end
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doc"""
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==(g::DirectProductGroupElem, h::DirectProductGroupElem)
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> Checks if two direct product group elements are the same.
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"""
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function (==)(g::DirectProductGroupElem, h::DirectProductGroupElem)
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g.elts == h.elts || return false
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return true
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end
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###############################################################################
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#
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# Group operations
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#
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###############################################################################
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doc"""
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*(g::DirectProductGroupElem, h::DirectProductGroupElem)
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> Return the direct-product group operation of elements, i.e. component-wise
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> operation as defined by `operations` field of the parent object.
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"""
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function *(g::DirectProductGroupElem{T}, h::DirectProductGroupElem{T}, check::Bool=true) where {T}
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if check
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parent(g) == parent(h) || throw(DomainError(
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"Can not multiply elements of different groups!"))
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end
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return DirectProductGroupElem([a*b for (a,b) in zip(g.elts,h.elts)])
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end
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doc"""
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inv(g::DirectProductGroupElem)
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> Return the inverse of the given element in the direct product group.
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"""
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function inv(g::DirectProductGroupElem{T}) where {T<:GroupElem}
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return DirectProductGroupElem([inv(a) for a in g.elts])
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end
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###############################################################################
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#
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# Misc
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#
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###############################################################################
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import Base: size, length, start, next, done, eltype
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struct DirectPowerIter{Gr<:AbstractAlgebra.Group, GrEl<:AbstractAlgebra.GroupElem}
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G::Gr
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N::Int
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elts::Vector{GrEl}
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totalorder::Int
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orderG::Int
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end
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function DirectPowerIter(G::Gr, N::Integer) where {Gr<:AbstractAlgebra.Group}
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return DirectPowerIter{Gr, elem_type(G)}(
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G,
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N,
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vec(collect(elements(G))),
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Int(order(G))^N,
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Int(order(G))
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)
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end
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Base.size(DPIter::DirectPowerIter) = ntuple(i -> DPIter.orderG, DPIter.N)
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Base.length(DPIter::DirectPowerIter) = DPIter.totalorder
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Base.start(::DirectPowerIter) = 0
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function Base.next(DPIter::DirectPowerIter, state)
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idx = ind2sub(size(DPIter), state+1)
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return DirectProductGroupElem([DPIter.elts[i] for i in idx]), state+1
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end
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Base.done(DPIter::DirectPowerIter, state) = DPIter.totalorder <= state
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Base.eltype(::Type{DirectPowerIter{Gr, GrEl}}) where {Gr, GrEl} = DirectProductGroupElem{GrEl}
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doc"""
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elements(G::DirectProductGroup)
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> Returns `generator` that produces all elements of group `G` (provided that
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> `G.group` implements the `elements` method).
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"""
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elements(G::DirectProductGroup) = DirectPowerIter(G.group, G.n)
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doc"""
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order(G::DirectProductGroup)
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> Returns the order (number of elements) in the group.
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"""
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order(G::DirectProductGroup) = order(G.group)^G.n
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