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#[caption]"Comparing two factorizations of $(3\,2\,1)$"
#================================================================
#: For the proof of Hurwitz theorem (\autoref{th.Hurwitz})
#: Just check that the $\SG4$-valued monodromy factorizations%
#:   $((1\,2),(2\,3),(1\,2),(1\,2))$ and%
#:   $((1\,2),(2\,3),(2\,3),(2\,3))$%
#:   are strongly Hurwitz equivalent.
#: The equivalence is given by%
#: $"mu"=\Gb\Gs_3\Gb\Gs_3\1\in\BG4$, where $\Gb=\Gs_2\Gs_1^2\Gs_2$
#----------------------------------------------------------------
Reread("braid.txt");   #: (see \file{braid})
F := FreeGroup(4);;    #: The free group $\FG4$ to act on
B := FreeGroup(3);;    #: The braid group $\BG4$ (we do not need the relations)
gens := GeneratorsOfGroup(F);;
beta := B.2*B.1^2*B.2; mu := beta*B.3*beta*B.3^-1;  #: The braids $\Gb$ and $\mu$
free := Braid(mu, F, gens);   #: The action on $\FG4$ (see \kwd{Braid} in \file{braid})
old := [(1,2), (2,3), (1,2), (1,2)];  #: Original monodromy factorization
new := List(free, x -> MappedWord(x, gens, old));  #: New factorization
#================================================================
#[endinput]
