Getting unexpected minimal word representations for element from GAP

Question

I'm working with a permutation group generated by some specific generators:

r1 := (1,2,3,4,5,6);
r2 := (7,8,9,10,11,12);
s1 := (1,7,10,4);
s2 := (2,8,11,5);
s3 := (3,9,12,6);

I also have a set of elements of a subgroup that I wrote in terms of these:

f1 := r1*s1^(-1)*s2*r2^(-1);
f15 := r2^(2)*s1*s3^(2)*r1;

However, if I use GAP to give me the minimal representations of these two elements, it returns what I expected for f1, but not for f15.

In particular:

puzzle := Group([r1,r2,s1,s2,s3]);; Size(puzzle);
479001600
brk_2> hom := EpimorphismFromFreeGroup(puzzle: names:=["r1","r2","s1","s2","s3"]);;
brk_2> PreImagesRepresentative(hom, f1);
r1*s1^-1*s2*r2^-1
brk_2> PreImagesRepresentative(hom, f15);
s3^2*r1^-1*s2*r1^2*s2*r1*s2^-1*(s1^2*r1^-2)^2*s1^2*r1

Why doesn't GAP return a word of length 6 for f15, since clearly one exists?

Reference: Here's the documentation that shows this method for generating minimal words.

Note: The word expressed does seem to be equal to f15:

brk_2> s3^2*r1^-1*s2*r1^2*s2*r1*s2^-1*(s1^2*r1^-2)^2*s1^2*r1;
(1,7)(2,3,12,8,5,6,9,11,10,4)
brk_2> f15;
(1,7)(2,3,12,8,5,6,9,11,10,4)

Answer 1

The function PreImagesRepresentative is not guaranteed to return a minimal word, and I see nothing in the GAP manual stating otherwise. The section you reference is technically correct but indeed somewhat misleading (and thus I will look into changing it): it is possible that in the cited example a shortest word was found - it just isn't guaranteed!

To actually get a minimal word, you can use the Factorization function. Continuing your example, we get this:

gap> Factorization(puzzle, f15);
r2^2*s3^2*s1*r1