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Dear Russell Blyth,

You asked about factoization of group elements in GAP4.

Unfortunately (as you mentioned already) the function `Factorization'

has disappeared in GAP4. This is now done by taking pre-images, but

the output of this can be rather ugly.

There are two possible solutions to this problem.

One is to convert the GAP3 code of Factorization to GAP4.

Another solution consists in constructing the map from the finite

group into the free group (rather than the other way round). Then we

can use the function `ImagesRepresentative'. This by itself does not

solve the problem, because the "principal" method for this

operation does basically the same as `PreImagesRepresentative'.

However, the "second" method is a lot better for your purpose.

Concretely:

gap> G:=SymmetricGroup(6); Sym( [ 1 .. 6 ] ) gap> r:=(1,2); s:=(1,2,3,4,5,6); (1,2) (1,2,3,4,5,6) gap> K:=Subgroup(G,[r,s]); Group([ (1,2), (1,2,3,4,5,6) ]) gap> F:=FreeGroup("x","y"); <free group on the generators [ x, y ]> gap> kk:= GeneratorsOfGroup(K); [ (1,2), (1,2,3,4,5,6) ] gap> ff:= GeneratorsOfGroup(F); [ x, y ] gap> hom:= GroupGeneralMappingByImages( K, F, kk, ff ); [ (1,2), (1,2,3,4,5,6) ] -> [ x, y ] gap> NiceImage:= ApplicableMethod(ImagesRepresentative,[hom,(2,3,4)],0,2); function( hom, elm ) ... end gap> NiceImage( hom, (2,3,4) ); y^4*x*y*x*y

As a general comment we would like to point out that the problem of

finding a short word (in the generators) for a group element is

an interesting one. However in GAP there is no clever

functionality for this as yet.

We hope this answer helps; if you have any more questions, please ask.

Best wishes,

Willem de Graaf

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