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Separation between Groups, FreeGroups and AutGroups
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AutGroups.jl
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64
AutGroups.jl
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@ -0,0 +1,64 @@
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module AutGroups
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using Groups
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using Permutations
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import Base: inv
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export IDSymbol, AutSymbol, AutWord
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export rmul_AutSymbol, lmul_AutSymbol, flip_AutSymbol, symmetric_AutSymbol
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immutable AutSymbol <: GSymbol
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gen::String
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pow::Int
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ex::Expr
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end
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IDSymbol(::Type{AutSymbol}) = AutSymbol("(id)", 0, :(IDAutomorphism(N)))
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change_pow(s::AutSymbol, n::Int) = reduce(AutSymbol(s.gen, n, s.ex))
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function inv(f::AutSymbol)
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symbol = f.ex.args[1]
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if symbol == :ɛ
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return change_pow(f, f.pow % 2)
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elseif symbol == :σ
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perm = invperm(f.ex.args[2])
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gen = string('σ', [Char(8320 + i) for i in perm]...)
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return AutSymbol(gen, f.pow, :(σ($perm)))
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elseif symbol == :(ϱ) || symbol == :λ
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return AutSymbol(f.gen, -f.pow, f.ex)
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elseif symbol == :IDAutomorphism
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return f
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else
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throw(ArgumentError("Don't know how to invert $f (of type $symbol)"))
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end
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end
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function rmul_AutSymbol(i,j, pow::Int=1)
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gen = string('ϱ',Char(8320+i), Char(8320+j)...)
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return AutSymbol(gen, pow, :(ϱ($i,$j)))
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end
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function lmul_AutSymbol(i,j, pow::Int=1)
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gen = string('λ',Char(8320+i), Char(8320+j)...)
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return AutSymbol(gen, pow, :(λ($i,$j)))
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end
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function flip_AutSymbol(j, pow::Int=1)
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gen = string('ɛ', Char(8320 + j))
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return AutSymbol(gen, pow%2, :(ɛ($j)))
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end
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function symmetric_AutSymbol(perm::Vector{Int}, pow::Int=1)
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perm = Permutation(perm)
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ord = order(perm)
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pow = pow % ord
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perm = perm^pow
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gen = string('σ', [Char(8320 + i) for i in array(perm)]...)
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return AutSymbol(gen, 1, :(σ($(array(perm)))))
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end
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typealias AutWord GWord{AutSymbol}
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end #end of module AutGroups
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25
FreeGroups.jl
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25
FreeGroups.jl
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@ -0,0 +1,25 @@
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module FreeGroups
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using Groups
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import Base: inv, convert
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export FGSymbol, IDSymbol
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immutable FGSymbol <: GSymbol
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gen::String
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pow::Int
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end
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IDSymbol(::Type{FGSymbol}) = FGSymbol("(id)", 0)
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FGSymbol(x::String) = FGSymbol(x,1)
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inv(s::FGSymbol) = FGSymbol(s.gen, -s.pow)
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convert(::Type{FGSymbol}, x::String) = FGSymbol(x)
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change_pow(s::FGSymbol, n::Int) = reduce(FGSymbol(s.gen, n))
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typealias FGWord GWord{FGSymbol}
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FGWord(s::FGSymbol) = FGWord([s])
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end #end of module FreeGroups
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153
Groups.jl
153
Groups.jl
@ -1,74 +1,33 @@
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module FreeGroups
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module Groups
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export GSymbol, AutSymbol, Word, GWord, FGWord, AutWord, FGAutomorphism
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export GSymbol, GWord
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export reduce!, reduce
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import Base: length, ==, hash, show
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import Base: one, inv, reduce, *, ^
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import Base: length, ==, hash, show, convert
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import Base: *, ^, convert
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import Base: one, inv, reduce, push!, unshift!
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abstract GSymbol
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immutable FGSymbol <: GSymbol
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gen::String
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pow::Int
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end
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immutable AutSymbol <: GSymbol
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gen::String
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pow::Int
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ex::Expr
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end
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IDSymbol(::Type{FGSymbol}) = FGSymbol("(id)", 0)
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IDSymbol(::Type{AutSymbol}) = AutSymbol("(id)", 0, :(IDAutomorphism(N)))
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FGSymbol(x::String) = FGSymbol(x,1)
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function show(io::IO, s::GSymbol)
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if s.pow == 1
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print(io, (s.gen))
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elseif s.pow == 0
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print(io, "(id)")
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if s.pow == 0 || s.pow == 1
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print(io, s.gen)
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else
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print(io, (s.gen)*"^$(s.pow)")
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end
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end
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(==)(s::GSymbol, t::GSymbol) = s.gen == t.gen && s.pow == t.pow
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length(s::GSymbol) = (s.pow == 0 ? 0 : 1)
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one{T<:GSymbol}(::Type{T}) = IDSymbol(T)
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one(s::GSymbol) = one(typeof(s))
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inv(s::FGSymbol) = FGSymbol(s.gen, -s.pow)
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convert(::Type{FGSymbol}, x::String) = FGSymbol(x)
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reduce(s::GSymbol) = (s.pow == 0 ? one(s) : s)
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change_pow(s::FGSymbol, n::Int) = reduce(FGSymbol(s.gen, n))
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change_pow(s::AutSymbol, n::Int) = reduce(AutSymbol(s.gen, n, s.ex))
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(^)(s::GSymbol, n::Integer) = change_pow(s, s.pow*n)
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function inv(f::AutSymbol)
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symbol = f.ex.args[1]
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if symbol == :ɛ
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return FreeGroups.change_pow(f, f.pow % 2)
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elseif symbol == :σ
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perm = invperm(f.ex.args[2])
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gen = string('σ', [Char(8320 + i) for i in perm]...)
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return AutSymbol(gen, f.pow, :(σ($perm)))
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elseif symbol == :(ϱ) || symbol == :λ
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return AutSymbol(f.gen, -f.pow, f.ex)
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elseif symbol == :IDAutomorphism
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return f
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else
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throw(ArgumentError("Don't know how to invert $f (of type $symbol)"))
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end
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end
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function (*){T<:GSymbol}(s::T, t::T)
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return GWord{T}([s])*t
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end
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(*){T<:GSymbol}(s::T, t::T) = return GWord{T}([s])*t
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abstract Word
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@ -107,11 +66,7 @@ immutable GWord{T<:GSymbol} <: Word
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symbols::Vector{T}
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end
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typealias FGWord GWord{FGSymbol}
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typealias AutWord GWord{AutSymbol}
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GWord{T<:GSymbol}(s::T) = GWord{T}([s])
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FGWord(s::FGSymbol) = FGWord([s])
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IDWord{T<:GSymbol}(::Type{T}) = GWord(one(T))
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IDWord{T<:GSymbol}(W::GWord{T}) = IDWord(T)
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@ -181,12 +136,9 @@ function show(io::IO, W::GWord)
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end
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end
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push!(W::GWord, x) = push!(W.symbols, x...)
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unshift!(W::GWord, x) = unshift!(W.symbols, x...)
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function r_multiply!(W::GWord, x; reduced::Bool=true)
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if length(x) > 0
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push!(W, x)
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push!(W.symbols, x...)
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end
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if reduced
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reduce!(W)
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@ -196,7 +148,7 @@ end
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function l_multiply!(W::GWord, x; reduced::Bool=true)
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if length(x) > 0
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unshift!(W, reverse(x))
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unshift!(W.symbols, reverse(x)...)
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end
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if reduced
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reduce!(W)
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@ -243,85 +195,4 @@ end
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(^)(x::GWord, n::Integer) = power_by_squaring(x,n)
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type FGAutomorphism{T<:GSymbol}
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domain::Vector{T}
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image::Vector{GWord{T}}
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map::Function
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function FGAutomorphism{T}(domain::Vector{T}, image::Vector{GWord{T}}, map::Function)
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length(domain) == length(unique(domain)) ||
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throw(ArgumentError("The elements of $domain are not unique"))
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length(domain) == length(image) ||
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throw(ArgumentError("Dimensions of image and domain must match"))
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# Set(vcat([[s.gen for s in reduce!(x).symbols]
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# for x in image]...)) == Set(s.gen for s in domain) ||
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# throw(ArgumentError("Are You sure that $image defines an automorphism??"))
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new(domain, image, map)
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end
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end
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function show(io::IO, X::FGAutomorphism)
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title = "Endomorphism of Free Group on $(length(X.domain)) generators, sending"
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map = ["$x ⟶ $y" for (x,y) in zip(X.domain, X.image)]
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join(io, vcat(title,map), "\n")
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end
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(==)(f::FGAutomorphism, g::FGAutomorphism) =
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f.domain == g.domain && f.image == g.image
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function aut_func_from_table(table::Vector{Tuple{Int,Int}}, GroupIdentity=one(FGWord))
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if length(table) == 0
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# warn("The map is not an automorphism")
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nothing
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end
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return v->reduce(*,GroupIdentity, v[idx]^power for (idx, power) in table)
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end
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function aut_func_from_word(domain, w::GWord)
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table = Vector{Tuple{Int, Int}}()
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for s in w.symbols
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pair = (findfirst([x.gen for x in domain], s.gen), s.pow)
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push!(table, pair)
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end
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return aut_func_from_table(table)
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end
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function FGMap(domain::Vector{FGSymbol}, image::Vector{GWord})
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function_vector = Vector{Function}()
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for word in image
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push!(function_vector, aut_func_from_word(domain, word))
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end
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return v -> Vector{FGWord}([f(v) for f in function_vector])
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end
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FGAutomorphism(domain::Vector{FGSymbol}, image::Vector{GWord}) =
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FGAutomorphism(domain, image, FGMap(domain, image))
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FGAutomorphism(domain::Vector{FGSymbol}, image::Vector{FGSymbol}) =
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FGAutomorphism(domain, Vector{GWord}(image))
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function FGAutomorphism(domain::Vector, image::Vector)
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FGAutomorphism(Vector{FGSymbol}(domain), Vector{GWord}(image))
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end
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function FGAutomorphism(domain, image)
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FGAutomorphism([domain...], [image...])
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end
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"""Computes the composition g∘f of two morphisms"""
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function compose(f::FGAutomorphism, g::FGAutomorphism)
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if length(f.image) != length(g.domain)
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throw(ArgumentError("Cannot compose $f and $g"))
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else
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h(v) = g.map(f.map(v))
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return FGAutomorphism(f.domain, h(f.domain), h)
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end
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end
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(*)(f::FGAutomorphism, g::FGAutomorphism) = compose(f,g)
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end
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