[Gap Forum] Group Hom. and Algebra Hom.

[Thomas Breuer]

Dear GAP Forum,

Alper Odabas wrote

> 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.
> [...]
> 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

There is indeed a problem with linear mappings whose image is trivial.
(Also the analogous problem exists for general linear mappings whose
preimage is trivial.)

This bug will be fixed with the next GAP update.
For the moment, the bug can be fixed by reading the file

    http://www.math.rwth-aachen.de/~Thomas.Breuer/gapfix/fix_vspchom_4_4_9

into the GAP session.

Thanks very much for the bug report.

All the best,
Thomas Breuer