# [GAP Forum] reza orfi from iran

Alexander Hulpke hulpke at math.colostate.edu
Wed Mar 31 23:45:32 BST 2004

```Dear Gap-Froum,

Reza Orfi wrote:

> I need all normalsubgroups of this group
> g=<x,y! x^3=y^3,(xy)^3xy^-1xyx^-1yxy^-1>
> And my computer can not list all
> normalsubgroups of g .

Usually these types of calculations are easiest done in an isomorphic
permutation group:

gap> f:=FreeGroup("x","y");x:=f.1;y:=f.2;
<free group on the generators [ x, y ]>
x
y
gap> n:=[x^3,y^3,(x*y)^3*x/y*x*y/x*y*x/y];
[ x^3, y^3, x*y*x*y*x*y*x*y^-1*x*y*x^-1*y*x*y^-1 ]
gap> g:=f/n;
<fp group on the generators [ x, y ]>
gap> hom:=IsomorphismPermGroup(g);;
gap> h:=Image(hom);
<permutation group with 2 generators>

# now h is a permutation group with the same structure

gap> Size(h);
9000
gap> l:=NormalSubgroups(h);
[ Group(()), <permutation group with 2 generators>,
<permutation group with 3 generators>, <permutation group with 2 generators>
, <permutation group of size 3 with 1 generators>,
<permutation group of size 75 with 15 generators>,
<permutation group of size 150 with 16 generators>,
<permutation group of size 9000 with 2 generators> ]
gap> l:=List(l,i->PreImage(hom,i));
[ Group(<fp, no generators known>), Group(<fp, no generators known>),
Group(<fp, no generators known>), Group(<fp, no generators known>),
Group(<fp, no generators known>), Group(<fp, no generators known>),
Group(<fp, no generators known>), Group(<fp, no generators known>) ]

Calculating concrete generators in the finitely presented group is a bit
more memory intensive:

gap> List(l,GeneratorsOfGroup);
[],
[y*x*y*x*y^-1*x*y^-1*x^-1*y^-1*x^-1*y*x^-1,
y*x*y*x^-1*y*x^-1*y^-1*x^-1*y^-1*x*y^-1*x],
[y*x*y^-1*x*y*x*y^-1*x,y*x^-1*y^-1*x^-1*y*x^-1*y^-1*x^-1,
y^-1*x^-1*y*x^-1*y^-1*x^-1*y*x^-1,y*x^-1*y*x^-1*y^-1*x^-1*y*x^-1*y],
[x,y*x*y^-1,y^-1*x*y],
[y*x*y*x^-1*y*x^-1*y*x^-1*y*x*y*x^-1*y^-1*x^-1],
[y*x*y*x*y*x^-1*y^-1*x^-1*y^-1*x^-1,y*x*y*x*y^-1*x^-1*y^-1*x^-1*y^-1*x,
y*x*y*x^-1*y^-1*x^-1*y^-1*x^-1*y*x,y*x*y^-1*x^-1*y^-1*x^-1*y^-1*x*y*x],
[y*x*y*x^-1*y*x*y*x^-1,y*x*y^-1*x*y*x*y^-1*x,
y*x^-1*y^-1*x^-1*y*x^-1*y^-1*x^-1],
[x,y]]

Best wishes,

Alexander Hulpke

-- Colorado State University, Department of Mathematics,
Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA
email: hulpke at math.colostate.edu, Phone: ++1-970-4914288
http://www.math.colostate.edu/~hulpke

```