[GAP Forum] quotient group homomorphisms

[Max Horn]

Dear Walter,

On 23.01.2014, at 16:49, Walter Becker <w_becker at hotmail.com> wrote:

> The question here is how to display the mapping (homomorphism)
> 
> h:G---->G/N[i] 
> 
> on the generators of the group G. For a specific case try this:
> 
> 
> 
> G is Ho,(C_4 X C_2)  this is group number 259  in the Hall-Senior
> 
> Tables and group number 138 in the Small group Library.

It is not clear to me what N[i] is in your general description and/or in your specific example. Some normal subgroup, I assume, but which?

> A specific presentation here is 
> 
> f:=FreeGroup("a", "b", "c"  );
> g:=f/[f.1^2,
>      f.2^2,
>      f.3^2,
>      (f.1^-1*f.2^-1*f.1*f.2)^2,
>      (f.1^-1*f.3^-1*f.1*f.3)^2,
>       (f.2^-1*f.3^-1*f.2*f.3),
> (f.1^-1*f.2^-1*f.1*f.2)^-1*f.3^-1*(f.1^-1*f.2^-1*f.1*f.2)*f.3*
> ((f.1^-1*f.3^-1*f.1*f.3)^-1*f.2^-1*(f.1^-1*f.3^-1*f.1*f.3)*f.2)^-1   
> ];
> 
> 
> 
> The subgroups of most interest here are (C_2 X C_2)wr C_2
> 
> of order 32 of which there are 3 cases.

Do you mean that these are the normal subgroups you want to factor out?

Anyway, I don't know which subgroups you mean exactly. But for the sake of argument, let's just take any. E.g. the normal closure of the subgroup generated by the first gneerator:

gap> n:=NormalClosure(g, Subgroup(g,[g.1]));
Group(<fp, no generators known>)
gap> Size(n);
16


> Most specivically what are the images 
> 
> h(f.1), h(f.2)  and h(f.3).

This can be done as follows. First, you need to obtain the quotient map h:

gap> h:=NaturalHomomorphismByNormalSubgroup(g,n);
[ a, b, c ] -> [ (), (1,2)(3,4), (1,3)(2,4) ]

Then you can use it to compute the images of the generators:

gap> Image(h, g.1);
()
gap> Image(h, g.2);
(1,2)(3,4)
gap> Image(h, g.3);
(1,3)(2,4)

# There is also a shortcut notation:
gap> g.3^h;
(1,3)(2,4)



Hope that helps,
Max