[GAP Forum] Composition of group homorphisms

[Alexander Hulpke]

Dear GAP-Forum,

Jose Joao Morais wrote:

>     How can I compose group homomorphisms?
>
>     I tried this code:
>
> gap> g := SymmetricGroup(5);;
> gap> h := AlternatingGroup(5);;
> gap> f1 := NaturalHomomorphismByNormalSubgroup(g,h);;
> gap> f2 := NaturalHomomorphismByNormalSubgroup(h,h);;
> gap> CompositionMapping(f1,f2);

The first issue is order - CompositionMapping takes homomorphisms in  
reversed order. So to first apply f1, the f2, you would use either
f1*f2
or, equivalently
CompositionMapping(f2,f1)

The second issue is domains and the fact that GAP is really picky  
about definitions which a mathematician intuitively would interpret  
in the only possible way.
Your homomorphism f2 is defined on h, and not on the factor group g/h.

What you could do, is either create this homomorphism in the factor  
group:

f3:=NaturalHomomorphismByNormalSubgroup(Image(f1,g),Image(f1,h));

Then you can form:
gap> CompositionMapping(f3,f1);
[ (1,2,3,4,5), (1,2) ] -> [ <identity> of ..., f1 ]

Note that I cannot compose with
NaturalHomomorphismByNormalSubgroup(Image(f1,h),Image(f1,h));
(the first `Image' changed from g to h), as then the image of the  
first homomorphism is not contained in the source of the second. You  
could use `RestrictedMapping' to deliberately restrict the domain of  
f1 to obtain a smaller image.

I hope this is of help,

    Alexander Hulpke




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