Add DirectProducts
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DirectProducts.jl
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248
DirectProducts.jl
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module DirectProducts
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using Nemo
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import Base: show, ==, hash, deepcopy_internal
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import Base: ×, *, inv
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import Nemo: parent, parent_type, elem_type
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import Nemo: elements, order, Group, GroupElem, Ring
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export DirectProductGroup, DirectProductGroupElem
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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(factors::Vector{Group}) <: Group
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Implements direct product of groups as vector factors. 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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type DirectProductGroup <: Group
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factors::Vector{Group}
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operations::Vector{Function}
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end
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type DirectProductGroupElem <: GroupElem
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elts::Vector{GroupElem}
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parent::DirectProductGroup
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DirectProductGroupElem{T<:GroupElem}(a::Vector{T}) = new(a)
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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(G::DirectProductGroup) = DirectProductGroupElem
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parent_type(::Type{DirectProductGroupElem}) = DirectProductGroup
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parent(g::DirectProductGroupElem) = g.parent
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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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DirectProductGroup(G::Group, H::Group) = DirectProductGroup([G, H], Function[(*),(*)])
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DirectProductGroup(G::Group, H::Ring) = DirectProductGroup([G, H], Function[(*),(+)])
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DirectProductGroup(G::Ring, H::Group) = DirectProductGroup([G, H], Function[(+),(*)])
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DirectProductGroup(G::Ring, H::Ring) = DirectProductGroup([G, H], Function[(+),(+)])
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DirectProductGroup{T<:Ring}(X::Vector{T}) = DirectProductGroup(Group[X...], Function[(+) for _ in X])
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×(G::Group, H::Group) = DirectProductGroup(G,H)
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function DirectProductGroup{T<:Group, S<:Group}(G::Tuple{T, Function}, H::Tuple{S, Function})
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return DirectProductGroup([G[1], H[1]], Function[G[2],H[2]])
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end
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function DirectProductGroup(groups::Vector)
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for G in groups
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typeof(G) <: Group || throw("$G is not a group!")
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end
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ops = Function[typeof(G) <: Ring ? (+) : (*) for G in groups]
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return DirectProductGroup(groups, ops)
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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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(G::DirectProductGroup)() = G([H() for H in G.factors]; checked=false)
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function (G::DirectProductGroup)(g::DirectProductGroupElem; checked=true)
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if checked
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return G(g.elts)
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else
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g.parent = G
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return g
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end
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end
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doc"""
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(G::DirectProductGroup)(a::Vector; checked=true)
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> Constructs element of the direct product group `G` by coercing each element
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> of vector `a` to the corresponding factor of `G`. If `checked` flag is set to
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> `false` no checks on the correctness are performed.
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"""
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function (G::DirectProductGroup)(a::Vector; checked=true)
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length(a) == length(G.factors) || throw("Cannot coerce $a to $G: they have
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different number of factors")
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if checked
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for (F,g) in zip(G.factors, a)
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try
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F(g)
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catch
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throw("Cannot coerce to $G: $g cannot be coerced to $F.")
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end
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end
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end
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elt = DirectProductGroupElem([F(g) for (F,g) in zip(G.factors, a)])
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elt.parent = G
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return elt
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end
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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 deepcopy_internal(g::DirectProductGroupElem, dict::ObjectIdDict)
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G = parent(g)
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return G(deepcopy(g.elts))
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end
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function hash(G::DirectProductGroup, h::UInt)
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return hash(G.factors, hash(G.operations, 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(g.parent, hash(DirectProductGroupElem, h)))
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end
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doc"""
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eye(G::DirectProductGroup)
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> Return the identity element for the given direct product of groups.
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"""
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eye(G::DirectProductGroup) = G()
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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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println(io, "Direct product of groups")
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join(io, G.factors, ", ", " and ")
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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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function (==)(G::DirectProductGroup, H::DirectProductGroup)
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G.factors == H.factors || return false
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G.operations == H.operations || 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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> Return `true` if the given elements of direct products are equal, otherwise return `false`.
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"""
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function (==)(g::DirectProductGroupElem, h::DirectProductGroupElem)
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parent(g) == parent(h) || return false
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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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# Binary operators
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#
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###############################################################################
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function direct_mult(g::DirectProductGroupElem, h::DirectProductGroupElem)
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parent(g) == parent(h) || throw("Can't multiply elements from different groups: $g, $h")
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G = parent(g)
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return G([op(a,b) for (op,a,b) in zip(G.operations, g.elts, h.elts)])
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end
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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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(*)(g::DirectProductGroupElem, h::DirectProductGroupElem) = direct_mult(g,h)
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###############################################################################
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#
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# Inversion
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#
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###############################################################################
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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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# TODO: dirty hack around `+` operation
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function inv(g::DirectProductGroupElem)
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G = parent(g)
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return G([(op == (*) ? inv(elt): -elt) for (op,elt) in zip(G.operations, 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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doc"""
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elements(G::DirectProductGroup)
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> Returns `Task` that produces all elements of group `G` (provided that factors
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> implement the elements function).
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"""
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# TODO: can Base.product handle generators?
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# now it returns nothing's so we have to collect ellements...
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function elements(G::DirectProductGroup)
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cartesian_prod = Base.product([collect(elements(H)) for H in G.factors]...)
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return (G(collect(elt)) for elt in cartesian_prod)
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end
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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) = prod([order(H) for H in G.factors])
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end # of module DirectProduct
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