[GAP Forum] Define function with domain and codomain

[Jack Schmidt]

GroupHomomorphismByImages or GroupHomomorphismByFunction are probably 
what you are looking for.  The first argument is the domain, and the 
second argument is the codomain.

Here is an example:

C4 := CyclicGroup(IsPermGroup,4);
S4 := SymmetricGroup(4);
dom := DirectProduct( C4, C4, C4, S4 );;
cod := GL(1,Integers);;
fun := h -> Product( [1..4], i -> SignPerm( Image( Projection( dom, i), 
h ) ) );;
homf := GroupHomomorphismByFunction( dom, cod, h -> [[ fun(h) ]] );;
homi := GroupHomomorphismByImages( dom, cod, GeneratorsOfGroup(dom), 
List( GeneratorsOfGroup(dom), h -> [[ fun(h) ]] ) );;
kerf := Kernel(homf);
keri := Kernel(homi);

gap> kerf=keri;
true
gap> StructureDescription(keri);
"C4 x C4 x (A4 : C4)"
gap> Elements(cod);
[ [ [ -1 ] ], [ [ 1 ] ] ]


Note that kerf has many more generators than keri, so it is less 
efficient to work with kerf than with keri.  However, as you can see, 
the groups are equal.

You cannot literally use [1,-1] as a group, since exponentiation is 
defined differently for group elements than for rational numbers, but 
you can use 1x1 matrices instead, which are equivalent in an easy to see 
way.

Inneke Van Gelder wrote:
> Dear GAP-forum,
> 
>  
> 
>  
> 
> How can I define a function with explicit domain and codomain? 
> 
> I need it do find the size of the kernel of the mapping
> 
> t: C_4^3 \times S_4 \rightarrow \{ \pm 1 \}: (h_1,h_2,h_3,h_4) \mapsto
> sgn(h_1)sgn(h_2)sgn(h_3)sgn(h_4)
> 
>  
> 
>  
> 
>  
> 
> Best regards,
> 
> Inneke Van Gelder
> 
>  
> 
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