Dear Gary McConnell, Dear Forum,
> I am new to GAP - I am using it inside SAGE ... I am trying to do a bunch of > computations involving counting how certain elements of S6 behave under > "the" outer automorphism. I have tried to do something from the manual pages > If say I had a list of elements of S6, entered in the usual way as cycles, > and if I chose a section s from Out(G) back to Aut(G), how would I get the > image s(rho) of rho (if I write Out(G)=<1,rho> say) under that section, > and/or how would I get the image of my elements under s(rho)? This is how you would do it in GAP: gap> G:=SymmetricGroup(6); Sym( [ 1 .. 6 ] ) gap> A:=AutomorphismGroup(G); <group with 3 generators> Now, in your case you just want a generator of G outside the inner automorphisms: gap> inn:=InnerAutomorphismsAutomorphismGroup(A); <group of size 720 with 2 generators> gap> s:=First(GeneratorsOfGroup(A),x->not x in inn); [ (5,6), (1,2,3,4,5) ] -> [ (1,2)(3,5)(4,6), (1,2,3,4,5) ] To calculate element images: gap> Image(s,(1,2,3)); (1,4,3)(2,6,5) Here s is probably not the nicest possible gap> Order(s); 10 Lets see what we can get: gap> Collected(List(Elements(inn),x->Order(x*s))); [ [ 2, 36 ], [ 4, 180 ], [ 8, 360 ], [ 10, 144 ] ] So order 2 for an outer automorphism is possible. To find one of this kind either: gap> news:=First(Elements(inn),x->Order(x*s)=2)*s; [ (1,2,3,4,5,6), (1,2) ] -> [ (2,6)(3,5,4), (1,2)(3,4)(5,6) ] gap> Order(news); 2 gap> news in inn; false or (less memory use, but more complicated -- use classes of A and find a representative, do so via permrep): gap> hom:=IsomorphismPermGroup(A); MappingByFunction( <group with 3 generators>, Group( [ (1,2,3,4,5,6)(7,12,8)(9,11), (1,2)(7,8)(9,10)(11,12), (1,7,4,10,2,8,5,11,3,9)(6,12) ]), function( auto ) ... end, function( perm ) ... end ) gap> AP:=Image(hom); Group([ (1,2,3,4,5,6)(7,12,8)(9,11), (1,2)(7,8)(9,10)(11,12), (1,7,4,10,2,8,5,11,3,9)(6,12) ]) gap> innP:=Image(hom,inn); Group([ (1,2,3,4,5,6)(7,12,8)(9,11), (1,2)(7,8)(9,10)(11,12) ]) gap> cl:=ConjugacyClasses(AP);; gap> cl:=Filtered(cl,x->not Representative(x) in innP);;Length(cl); 5 gap> List(cl,x->[Order(Representative(x)),Size(x)]); [ [ 8, 180 ], [ 4, 180 ], [ 2, 36 ], [ 8, 180 ], [ 10, 144 ] ] gap> newsp:=Representative(cl[3]); (1,8)(2,10)(3,11)(4,7)(5,12)(6,9) gap> PreImagesRepresentative(hom,newsp); [ (1,2,3,4,5,6), (1,2) ] -> [ (1,4,5)(3,6), (1,4)(2,6)(3,5) ] I hope this helps. Best wishes, Alexander Hulpke -- Colorado State University, Department of Mathematics, Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA email: hul...@math.colostate.edu, Phone: ++1-970-4914288 http://www.math.colostate.edu/~hulpke _______________________________________________ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum