F. Harou writes:
I can reformulate more precisely my problem :
first, I have an exact sequence0 -> K -> G -> H -> 0 H is PSL(2,Z/7Z) so order(H) is 168. G is < x,y | x^-1 * y * x * y^-1 * x * y * x^-1 * y^-1 * x >
I try to find a presentation of K for computing the rank of K/[K,K].
I have a representation of G in PSL(2,O_3) and the homomorphism
G -> H for this presentation but it seem to give no more information because
the order of G is infinite. What could I do ?
The map you gave yesterday was onto SL(2,7), not PSL(2,7), and with the
given generators it does not appear to be a homomorphic image of G.
gap> un:=Z(7)^0; gap> zero:=0*Z(7); gap> gi1:=[[un, un], [zero, un]]; gap> gi2:=[[un, zero], [2*un, un]]; gap> x:=gi1;; y:=gi2;; gap> x^-1 * y * x * y^-1 * x * y * x^-1 * y^-1 * x; [ [ Z(7)^3, Z(7)^2 ], [ Z(7)^5, Z(7) ] ]
The value of the relator is not the identity in the image group.
If you tell me the correct images of x and y in G in PSL(2,7), then I can
show you how to calculate a presentation of the kernel K, and hence K/[K,K].
Roughly, you work out the map of G onto the regular permutation of PSL(2,7),
and use that to form a coset table for G, from which you can get a
presentation of the subgroup (which is the stabiliser of a point in the
regular permutation representation - i.e. K).