[GAP Forum] Quotient group

[Rudolf Zlabinger]

Dear D.Naidu,

i hope, the following helps:

Prolog

gap> a4:=AlternatingGroup(4);;S5:=SymmetricGroup(5);;
gap> g:=DirectProduct(a4,S5);;
gap> IsNormal(g,a4);  # lets a4 be our normal subgroup H, g our group G
true

Create the mapping and the image

gap> nhom:=NaturalHomomorphismByNormalSubgroup(g,a4);;
gap> image:=Image(nhom);; # G/H

Show the full  Preimage

gap> pre:=PreImages(nhom,image);  # this results in group G
Group([ (5,6,7,8,9), (5,6), (2,3,4), (1,2,3) ])
gap> g=pre;
true

x=p(u(x))

gap> rimage:=Random(image);; # choose a random elm of G/H, say our x
gap> pre:=PreImages(nhom,rimage); # this is our u(x), you have to use 
PreImages because the mapping is not unique by group elements
RightCoset(Group( [ (2,3,4), (1,2,3) ] ),(5,7)(6,8))
gap> repre:=Image(nhom,pre);; # this i a preform of p(u(x))
gap> repre=rimage;
false
gap> rimage;
(1,3)(2,4)
gap> repre;
[ (1,3)(2,4) ]                 # as homomorphisms in general are not unique 
in all cases, the algorithm provides for more than one value
gap> repre[1]=rimage;  # in this way repre[1] = p(u(x)) = x = rimage in our 
case of a quotient group homomorphism
true

If you want to do it for all elms of G/H, use listprocessing.

best wishes, Rudolf Zlabinger

----- Original Message ----- 
From: "D N" <dn2447 at yahoo.com>
To: <forum at gap-system.org>
Sent: Thursday, October 26, 2006 12:50 AM
Subject: [GAP Forum] Quotient group


> Dear GAP forum,
>
> Let H be a normal subgroup of a finite group G and
> let p: G --> H\G be the usual surjection.
> How do  I do the following using GAP:
>
> * form the quotient group H\G
> * for each x in H\G, chose u(x) in G
>  such that p(u(x)) = x.
>
> Thanks,
> D. Naidu
>
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