Recently I saw a post at rec.puzzles regarding a permutation puzzle, which
the poster had done some analysis using magma. I wanted to do the same sort
of analysis with GAP.
Here's my take on it:
gap> f := FreeGroup(["a","b","c","d"]);; gap> a := f.1;; b:= f.2;; c:= f.3;; d := f.4;; gap> pgens := [(1,2,3,7,11,10,9,5), (2,3,4,8,12,11,10,6), (5,6,7,11,15,14,13,9), (6,7,8,12,16,15,14,10)];; gap> puzzle := Group(pgens);; gap> h := GroupHomomorphismByImages(f, puzzle, [a,b,c,d], pgens); [ a, b, c, d ] -> [ (1,2,3,7,11,10,9,5), (2,3,4,8,12,11,10,6), (5,6,7,11,15,14,13,9), (6,7,8,12,16,15,14,10) ] gap> gword := PreImagesRepresentative(h, (15,16));;
But I'm not absolutely certain this is the shortest word:)
gap> Length(gword); 796257
Perhaps someone else knows how to find a shorter one.
p.s I'm very curious to know what the puzzle is.
p.p.s The generators you give for the puzzle generate the symmetric group
on the 16 points.