How do i see how all the generators get mapped during automorphism in GAP

Question

I'm using GAP for the first time. I used AllAutomorphism command to see all automorphisms of SymmetricGroup of order $3$.

gap> AllAutomorphisms(SymmetricGroup(3));

     [ ^(), ^(2,3), ^(1,3), ^(1,3,2), ^(1,2,3), ^(1,2) ]

I want to understand what the output means? I would also like to see how generators(or group elements) map during each automorphism map.

Answer 1

The notation ^perm denotes an inner automorphism -- conjugation by the indicated permutation. The result tells you that $S_3$ has no outer automorphisms. If you want to see the action on generators, you could use AsGroupGeneralMappingByImages to convert the representation:

gap> all:=AllAutomorphisms(SymmetricGroup(3));
[ ^(), ^(2,3), ^(1,3), ^(1,3,2), ^(1,2,3), ^(1,2) ]
gap> List(all,AsGroupGeneralMappingByImages);
[ [ (1,2,3), (1,2) ] -> [ (1,2,3), (1,2) ],
  [ (1,2,3), (1,2) ] -> [ (1,3,2), (1,3) ],
  [ (1,2,3), (1,2) ] -> [ (1,3,2), (2,3) ],
  [ (1,2,3), (1,2) ] -> [ (1,2,3), (1,3) ],
  [ (1,2,3), (1,2) ] -> [ (1,2,3), (2,3) ],
  [ (1,2,3), (1,2) ] -> [ (1,3,2), (1,2) ] ]

If you want to see what happens to group elements, you can apply the automorphisms with `Image'. For example

gap> elms:=Elements(SymmetricGroup(3));
[ (), (2,3), (1,2), (1,2,3), (1,3,2), (1,3) ]
gap> List(all,x->List(elms,y->Image(x,y)));
[ [ (), (2,3), (1,2), (1,2,3), (1,3,2), (1,3) ],
  [ (), (2,3), (1,3), (1,3,2), (1,2,3), (1,2) ],
  [ (), (1,2), (2,3), (1,3,2), (1,2,3), (1,3) ],
  [ (), (1,2), (1,3), (1,2,3), (1,3,2), (2,3) ],
  [ (), (1,3), (2,3), (1,2,3), (1,3,2), (1,2) ],
  [ (), (1,3), (1,2), (1,3,2), (1,2,3), (2,3) ] ]