Direct Product of Groups

DirectProductGroups.jl

module DirectProduct

using Nemo

import Base: show, ==, hash
import Base: ×, *, inv

import Nemo: parent, parent_type, elem_type
import Nemo: elements, order, Group, GroupElem, Ring

export DirectProductGroup, DirectProductGroupElem

###############################################################################
#
#   DirectProductGroup / DirectProductGroupElem
#
###############################################################################

doc"""
   DirectProductGroup(factors::Vector{Group}) <: Group
Implements direct product of groups as vector factors. The group operation is
`*` distributed component-wise, with component-wise identity as neutral element.
"""

type DirectProductGroup <: Group
   factors::Vector{Group}
   operations::Vector{Function}
end

type DirectProductGroupElem <: GroupElem
   elts::Vector{GroupElem}
   parent::DirectProductGroup

   DirectProductGroupElem{T<:GroupElem}(a::Vector{T}) = new(a)
end

###############################################################################
#
#   Type and parent object methods
#
###############################################################################

elem_type(G::DirectProductGroup) = DirectProductGroupElem

parent_type(::Type{DirectProductGroupElem}) = DirectProductGroup

parent(g::DirectProductGroupElem) = g.parent

###############################################################################
#
#   DirectProductGroup / DirectProductGroupElem constructors
#
###############################################################################

DirectProductGroup(G::Group, H::Group) = DirectProductGroup([G, H], Function[(*),(*)])

DirectProductGroup(G::Group, H::Ring) = DirectProductGroup([G, H], Function[(*),(+)])

DirectProductGroup(G::Ring, H::Group) = DirectProductGroup([G, H], Function[(+),(*)])

DirectProductGroup(G::Ring, H::Ring) = DirectProductGroup([G, H], Function[(+),(+)])

DirectProductGroup{T<:Ring}(X::Vector{T}) = DirectProductGroup(Group[X...], Function[(+) for _ in X])

×(G::Group, H::Group) = DirectProductGroup(G,H)

function DirectProductGroup{T<:Group, S<:Group}(G::Tuple{T, Function}, H::Tuple{S, Function})
   return DirectProductGroup([G[1], H[1]], Function[G[2],H[2]])
end

function DirectProductGroup(groups::Vector)
   for G in groups
      typeof(G) <: Group || throw("$G is not a group!")
   end
   ops = Function[typeof(G) <: Ring ? (+) : (*) for G in groups]

   return DirectProductGroup(groups, ops)
end

###############################################################################
#
#   Parent object call overloads
#
###############################################################################

(G::DirectProductGroup)() = G([H() for H in G.factors]; checked=false)

function (G::DirectProductGroup)(g::DirectProductGroupElem; checked=true)
   if checked
      return G(g.elts)
   else
      g.parent = G
      return g
   end
end

doc"""
    (G::DirectProductGroup)(a::Vector; checked=true)
> Constructs element of the direct product group `G` by coercing each element
> of vector `a` to the corresponding factor of `G`. If `checked` flag is set to
> `false` no checks on the correctness are performed.

"""
function (G::DirectProductGroup)(a::Vector; checked=true)
   length(a) == length(G.factors) || throw("Cannot coerce $a to $G: they have
      different number of factors")
   if checked
      for (F,g) in zip(G.factors, a)
         try
            F(g)
         catch
            throw("Cannot coerce to $G: $g cannot be coerced to $F.")
         end
      end
   end
   elt = DirectProductGroupElem([F(g) for (F,g) in zip(G.factors, a)])
   elt.parent = G
   return elt
end

###############################################################################
#
#   Basic manipulation
#
###############################################################################

function hash(G::DirectProductGroup, h::UInt)
   return hash(G.factors, hash(G.operations, hash(DirectProductGroup,h)))
end

function hash(g::DirectProductGroupElem, h::UInt)
   return hash(g.elts, hash(g.parent, hash(DirectProductGroupElem, h)))
end

doc"""
    eye(G::DirectProductGroup)
> Return the identity element for the given direct product of groups.

"""
eye(G::DirectProductGroup) = G()

###############################################################################
#
#   String I/O
#
###############################################################################

function show(io::IO, G::DirectProductGroup)
    println(io, "Direct product of groups")
    join(io, G.factors, ", ", " and ")
end

function show(io::IO, g::DirectProductGroupElem)
    print(io, "("*join(g.elts,",")*")")
end

###############################################################################
#
#   Comparison
#
###############################################################################

function (==)(G::DirectProductGroup, H::DirectProductGroup)
    G.factors == H.factors || return false
    G.operations == H.operations || return false
    return true
end

doc"""
    ==(g::DirectProductGroupElem, h::DirectProductGroupElem)
> Return `true` if the given elements of direct products are equal, otherwise return `false`.

"""
function (==)(g::DirectProductGroupElem, h::DirectProductGroupElem)
    parent(g) == parent(h) || return false
    g.elts == h.elts || return false
    return true
end

###############################################################################
#
#   Binary operators
#
###############################################################################

function direct_mult(g::DirectProductGroupElem, h::DirectProductGroupElem)
    parent(g) == parent(h) || throw("Can't multiply elements from different groups: $g, $h")
    G = parent(g)
    return G([op(a,b)  for (op,a,b) in zip(G.operations, g.elts, h.elts)])
end

doc"""
    *(g::DirectProductGroupElem, h::DirectProductGroupElem)
> Return the direct-product group operation of elements, i.e. component-wise
> operation as defined by `operations` field of the parent object.

"""
(*)(g::DirectProductGroupElem, h::DirectProductGroupElem) = direct_mult(g,h)

###############################################################################
#
#   Inversion
#
###############################################################################

doc"""
    inv(g::DirectProductGroupElem)
> Return the inverse of the given element in the direct product group.

"""
# TODO: dirty hack around `+` operation
function inv(g::DirectProductGroupElem)
    G = parent(g)
    return G([(op == (*) ? inv(elt): -elt) for (op,elt) in zip(G.operations, g.elts)])
end

###############################################################################
#
#   Misc
#
###############################################################################

doc"""
    elements(G::DirectProductGroup)
> Returns `Task` that produces all elements of group `G` (provided that factors
> implement the elements function).

"""
# TODO: can Base.product handle generators?
#   now it returns nothing's so we have to collect ellements...
function elements(G::DirectProductGroup)
    cartesian_prod = Base.product([collect(elements(H)) for H in G.factors]...)

    function _it()
        for elt in cartesian_prod
            produce(G(collect(elt)))
        end
    end

    return Task(_it)
end

doc"""
    order(G::DirectProductGroup)
> Returns the order (number of elements) in the group.

"""
order(G::DirectProductGroup) = prod([order(H) for H in G.factors])

end # of module DirectProduct