[Gap Forum] Group Hom. and Algebra Hom.

[Alper Odabaş]

Let f:A -> A homomorphism with f(a)=0, g:A -> {0} with g(a)=0 and h:{0} -> A with h(0)=0. (A is a Algebra ) 



there is a error (algebra case) with these functions in GAP 4.4.7.







ALGEBRA CASE ---- not working

 

gap> A:=GroupRing(GF(3),CyclicGroup(2));
<algebra-with-one over GF(3), with 1 generators>
gap> gA:=GeneratorsOfAlgebra(A);
[ (Z(3)^0)*<identity> of ..., (Z(3)^0)*f1 ]
gap> zA:=Zero(A);;
gap> im:=[zA,zA];;
gap> f:=AlgebraHomomorphismByImages(A,A,gA,im);;
gap> IsSurjective(f);
false
gap> IsInjective(f);
Error, no method found! For debugging hints type ?Recovery from NoMethodFound
Error, no 1st choice method found for `SemiEchelonMatTransformation' on 1 argu\
ments called from
SemiEchelonMatTransformation( List( mapi[2], function ( x )
        return Coefficients( B, x );
    end ) ) called from
MakePreImagesInfoLinearGeneralMappingByImages( map ); called from
IsInjective( AsLeftModuleGeneralMappingByImages( map ) ) called from
<function>( <arguments> ) called from read-eval-loop
Entering break read-eval-print loop ...
you can 'quit;' to quit to outer loop, or
you can 'return;' to continue
brk> 
gap> B:=Image(f);
<algebra of dimension 0 over GF(3)>
gap> g:=AlgebraHomomorphismByImages(A,B,gA,im);;
gap> IsSurjective(g);
true
gap> IsInjective(g);
same error with above
gap> gB:=GeneratorsOfAlgebra(B);
[ <zero> of ..., <zero> of ... ]
gap> h:=AlgebraHomomorphismByImages(B,A,gB,gB);
Error, no method found! For debugging hints type ?Recovery from NoMethodFound
Error, no 1st choice method found for `SemiEchelonMatTransformation' on 1 argu\
ments called from
SemiEchelonMatTransformation( List( mapi[1], function ( x )
        return Coefficients( B, x );
    end ) ) called from
MakeImagesInfoLinearGeneralMappingByImages( map ); called from
IsSingleValued( AsLeftModuleGeneralMappingByImages( map ) ) called from
IsMapping( hom ) called from
<function>( <arguments> ) called from read-eval-loop
Entering break read-eval-print loop ...
you can 'quit;' to quit to outer loop, or
you can 'return;' to continue
brk> 

 

GROUP CASE ---- no problem

 

gap> AA:=Group((1,2,3),(2,3));
Group([ (1,2,3), (2,3) ])
gap> gAA:=GeneratorsOfGroup(AA);
[ (1,2,3), (2,3) ]
gap> imm:=[(),()];
[ (), () ]
gap> ff:=GroupHomomorphismByImages(AA,AA,gAA,imm);
[ (1,2,3), (2,3) ] -> [ (), () ]
gap> IsSurjective(ff);
false
gap> IsInjective(ff);
false
gap> BB:=Image(ff);
Group(())
gap> g:=GroupHomomorphismByImages(AA,BB,gAA,imm);
[ (1,2,3), (2,3) ] -> [ (), () ]
gap> gg:=GroupHomomorphismByImages(AA,BB,gAA,imm);
[ (1,2,3), (2,3) ] -> [ (), () ]
gap> IsSurjective(gg);
true
gap> IsInjective(gg);
false
gap> gBB:=GeneratorsOfGroup(BB);
[ (), () ]
gap> hh:=GroupHomomorphismByImages(BB,AA,gBB,gBB);
[ (), () ] -> [ (), () ]
gap> IsSurjective(hh);
false
gap> IsInjective(hh);
true
gap>