[GAP Forum] quotient group homomorphisms
Max Horn
max at quendi.de
Tue Jan 28 00:33:26 GMT 2014
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
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