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Dear GAP forum,

Peter F. Blanchard wrote [in reference to my AutAg share package]:

>> Here is my question: can someone tell me how to multiply cosets

>> of subgroups of automorphism groups? It seems that this was left out, but

>> it is probably not difficult to do. An example follows which illustrates

>> how the problem comes up when trying to calculate an intersection of

>> subgroups of an automorphism group.

I think I need an explanation from someone who knows more about the parsing

of GAP commands. I'm replying to the list because this may be of interest

to others who are interested in implementing their own group element

records...

I've repeated Peter's example here (with a few more commands added):

---------------------------------------------------------------------- gap> G := SpecialAgGroup(DihedralGroup(AgWords,8)); Group( s, d1, d2 ) gap> G.name := "G";; gap> A := AutGroupSagGroup(G); Group( Aut(G, [ s*d1, d1, d2 ]), InnerAut(G, d1), InnerAut(G, s) ) gap> A.name := "A";; gap> t1 := Subgroup(A, [A.1]); Subgroup( A, [ Aut(G, [ s*d1, d1, d2 ]) ] ) gap> t2 := Subgroup(A, [A.2]); Subgroup( A, [ InnerAut(G, d1) ] ) gap> Intersection(t1, t2); Error, product of <a> and <b> is not defined in <rec1> * <rec2> called from arg[1].operations.Stabilizer( arg[1], arg[2], arg[3] ) called from Stabilizer( H, Coset( G ), OnRight ) called from D.operations.Intersection( I, D ) called from Intersection( t1, t2 ) called from main loop brk> a; (Subgroup( A, [ Aut(G, [ s*d1, d1, d2 ]) ] )*InnerAut(G, IdAgWord)) brk> IsRightCoset(a); true brk> a.operations; DomainOps brk> a.operations.\*; function ( C, D ) ... end brk> Print(a.operations.\*); function ( C, D ) local E; if IsRightCoset( C ) and D in Parent( C.group ) then E := RightCoset( C.group, C.representative * D ); elif IsRightCoset( C ) then E := Elements( C ) * D; elif IsRightCoset( D ) then E := C * Elements( D ); else Error( "product of <C> and <D> is not defined" ); fi; return E; endbrk> Parent(b.group); G brk> b in Parent(a.group); true brk> RightCoset( a.group, a.representative * b); (Subgroup( A, [ Aut(G, [ s*d1, d1, d2 ]) ] )*InnerAut(G, d1)) brk> brk> Print(b.operations.\*); function ( a, b ) local prd; if IsAut( a ) and IsAut( b ) then if AutOps.IsKnownInner( a ) and AutOps.IsKnownInner( b ) then prd := AutOps.MakeAut( a.group, a.inner * b.inner ); else prd := AutOps.MakeAut( a.group, AutOps.Compose( a, b ) ); if IsBound( a.inverse ) and IsBound( b.inverse ) then prd.inverse := AutOps.MakeAut( a.group, AutOps.Compose( b.inverse, a.inverse ) ); prd.inverse.inverse := prd; fi; fi; elif IsList( a ) then prd := List( a, function ( x ) return x * b; end ); elif IsList( b ) then prd := List( b, function ( x ) return a * x; end ); else Error( "product of <a> and <b> is not defined" ); fi; return prd; end ----------------------------------------------------------------------

I guess I have 2 questions:

(1) When asked to evaluate "a * b", why wasn't the "a.operations.\*"

function used first?

(2) How should I amend my "b.operations.\*" to cope with this situation?

Should I replace the last "else-error" clause with:

elif IsBound(a.operations.\*) then

return a.operations.\*(a,b);

else

Error( "product of <a> and <b> is not defined" );

fi;

This solution appears to work, and I'm happy to update my code, but it

would be useful to know the "correct" solution to this problem.

Cheers,

Michael.

----------------------------------------------------------------------------- Michael.Smith@maths.anu.edu.au DSTO & Australian National University

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