> < ^ Date: Thu, 17 Sep 1998 17:05:33 +0100
> ^ From: Russell Blyth <blythrd@SLU.EDU >
> ^ Subject: Factorization

Dear Gap Forum,

(This is a duplicate of a message sent 24 hours ago to Gap4 - I apologize
to readers who subscribe to both lists for the repetition - but perhaps a
reader of Gap Forum who does not subscribe to Gap4 can provide help.)

I am teaching a first abstract algebra course (for undergraduates), and
will use Gap in a project written by Loren Larson (on sliding block
puzzles). Last time I
used the project with gap3; now we have installed gap4. In gap3 one can use
the function Factorization to express an element of a group as a product of
group generators. This "black box" (from the point of view of the student)
function worked nicely on the small groups encountered in the project.

Factorization appears to have disappeared from gap4. It is possible to
obtain a "factorization" using the GroupHomomorphismsByImages and
PreImagesRepresentative functions.

Is there another work-around that is less cumbersome for the students (who
won't see homomorphisms for another week or two, and not to the depth
needed to explain what these commands are doing)?

Failing that, how does one use Gap to simplify/rewrite the output of the
PreImagesRepresentative function to obtain a "canonical" product. To wit:

```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> hom:=GroupHomomorphismByImagesNC(F,K,GeneratorsOfGroup(F),
> GeneratorsOfGroup(K));
[ x, y ] -> [ (1,2), (1,2,3,4,5,6) ]
gap> PreImagesRepresentative(hom,(2,3,4));
y^-4*x^-1*y^-2*x^-1*y^-1*x^-1*y^-5*x^-1*y^-1*x^-1*y^-4*x^-1*y^-6*x^-1*y^-2*x^
-1*y^-1*x^-1*y^-5*x^-1*y^-1*x^-1*y^-4*x^-1*y^-2*x^-1*y^-1*x^-1*y^-5*x^-1*y^
-1*x^-2*y^-1*x^-1*y^-1*x^-1*y^-4*x^-1*y^-1*x^-1*y^-1*x^-1*y^-4*x^-1*y^-2*x^
-1*y^-1*x^-1*y^-5*x^-1*y^-1*x^-1*y^-4*x^-1*y^-2*x^-1*y^-1*x^-1*y^-5*x^-1*y^
-1*x^-1*y^-4*x^-1*y^-6*x^-1*y^-2*x^-1*y^-1*x^-1*y^-5*x^-1*y^-1*x^-2*y^-1*x^
-1*y^-1*x^-1*y^-1*x^-1*y^-1
```

(I also tried using the -O option to reinstate some gap3 capabilities, but
this apparently does not include Factorization.)

Thanks!

Russell Blyth

```===========================================================================
Russell D. Blyth                                 blythrd@slu.edu
Associate Professor                              http://sylow.slu.edu
Department of Mathematics and Computer Science   Voice: (314)977-2458
Saint Louis University                           Department: (314)977-2444
221 N. Grand Blvd.                               Fax: (314)977-3649
St. Louis, MO 63103                              Dept: http://euler.slu.edu
USA                                              Univ: http://www.slu.edu
===========================================================================
```

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