78 lines
2.1 KiB
Julia
78 lines
2.1 KiB
Julia
module SpecialAutomorphisms
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using AbstractAlgebra
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using Groups
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import AbstractAlgebra.perm
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###############################################################################
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#
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# Generating set
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#
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###############################################################################
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function generatingset(N::Int)
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SAutFN = AutGroup(FreeGroup(N), special=true)
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S = gens(SAutFN);
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return SAutFN, unique([S; inv.(S)])
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end
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function generatingset(parsed_args)
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N = parsed_args["N"]
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return generatingset(N)
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end
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###############################################################################
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#
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# Action of WreathProductElems on AutGroupElem
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#
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###############################################################################
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function AutFG_emb(A::AutGroup, g::WreathProductElem)
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isa(A.objectGroup, FreeGroup) || throw("Not an Aut(Fₙ)")
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parent(g).P.n == length(A.objectGroup.gens) || throw("No natural embedding of $(parent(g)) into $A")
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powers = [(elt == parent(elt)() ? 0: 1) for elt in g.n.elts]
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elt = reduce(*, [A(Groups.flip_autsymbol(i))^pow for (i,pow) in enumerate(powers)])
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Groups.r_multiply!(elt, [Groups.perm_autsymbol(g.p)])
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return elt
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end
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function AutFG_emb(A::AutGroup, p::perm)
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isa(A.objectGroup, FreeGroup) || throw("Not an Aut(Fₙ)")
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parent(p).n == length(A.objectGroup.gens) || throw("No natural embedding of $(parent(g)) into $A")
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return A(Groups.perm_autsymbol(p))
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end
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function (g::WreathProductElem)(a::Groups.Automorphism)
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g = AutFG_emb(parent(a),g)
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return g*a*g^-1
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end
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function (p::perm)(a::Groups.Automorphism)
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g = AutFG_emb(parent(a),p)
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return g*a*g^-1
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end
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autS(N::Int) = WreathProduct(PermutationGroup(2), PermutationGroup(N))
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function autS(parsed_args)
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return autS(parsed_args["N"])
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end
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###############################################################################
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#
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# Misc
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#
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###############################################################################
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function groupname(parsed_args)
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N = parsed_args["N"]
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return groupname(N), N
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end
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groupname(N::Int) = "SAutF$(N)"
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end # of module SpecialAutomorphisms
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