GroupsWithPropertyT/groups/autfreegroup.jl

module SpecialAutomorphisms

using Nemo
using Groups

if VERSION >= v"0.6.0"
   import Nemo.Generic.perm
end

###############################################################################
#
#  Generating set
#
###############################################################################

function generatingset(N::Int)

   SAutFN = AutGroup(FreeGroup(N), special=true)
   S = gens(SAutFN);

   return SAutFN, unique([S; inv.(S)])
end

function generatingset(parsed_args)
   N = parsed_args["N"]
   return generatingset(N)
end

###############################################################################
#
#  Action of WreathProductElems on AutGroupElem
#
###############################################################################

function AutFG_emb(A::AutGroup, g::WreathProductElem)
    isa(A.objectGroup, FreeGroup) || throw("Not an Aut(Fₙ)")
    parent(g).P.n == length(A.objectGroup.gens) || throw("No natural embedding of $(parent(g)) into $A")
    powers = [(elt == parent(elt)() ? 0: 1) for elt in g.n.elts]
    elt = reduce(*, [A(Groups.flip_autsymbol(i))^pow for (i,pow) in enumerate(powers)])
    Groups.r_multiply!(elt, [Groups.perm_autsymbol(g.p)])
    return elt
end

function AutFG_emb(A::AutGroup, p::perm)
    isa(A.objectGroup, FreeGroup) || throw("Not an Aut(Fₙ)")
    parent(p).n == length(A.objectGroup.gens) || throw("No natural embedding of $(parent(g)) into $A")
    return A(Groups.perm_autsymbol(p))
end

function (g::WreathProductElem)(a::AutGroupElem)
    g = AutFG_emb(parent(a),g)
    return g*a*g^-1
end

function (p::perm)(a::AutGroupElem)
    g = AutFG_emb(parent(a),p)
    return g*a*g^-1
end

autS(N::Int) = WreathProduct(PermutationGroup(2), PermutationGroup(N))

function autS(parsed_args)
   return autS(parsed_args["N"])
end

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

function groupname(parsed_args)
   N = parsed_args["N"]
   return groupname(N), N
end

groupname(N::Int) = "SAutF$(N)"

end # of module SpecialAutomorphisms