Matrix representation for PGL(n,q) in Magma

Question

In Magma (Computational Algebra System), we can define both groups $GL(n,K)$ and $PGL(n,k)$. For instance:

$K:=GF(7)$;

$G:=GL(2,K)$;

$G1:=PGL(2,K)$;

Now, one can easily construct a subgroup $H$ of $G:=GL(2,K)$, generated by some matrix $M$, as follows:

$M:=G!Matrix(K,2,2,[1,2,3,4]);$

$H:=sub<G| M>;$

My question is: Why can't we make a similar construction using the projective case $G1:=PGL(2,K)$? When I try to replace $G$ above by $G1$, I always get an error message.

Any hint/suggestion is appreciated.

Answer 1

The elements of ${\rm PGL}(2,K)$ are not matrices, so it is not surprising that this construction does not work! Theoretically these elements should be cosets of the centre of ${\rm GL}(2,K)$ in ${\rm GL}(2,K)$, but in Magma ${\rm PGL}(2,K)$ is defined as a permutation group. To do what you want, you need to define the homomorphism from ${\rm GL}(2,K)$ to ${\rm PGL}(2,K)$, which you can do as follows.

> K := GF(7);
> G := GL(2,K);
> V := VectorSpace(G);
> phi, G1 := OrbitAction(G,sub<V|V.1>);
  // G1 is the image of phi, which is PGL(2,K)
> M := G!Matrix(K,2,2,[1,2,3,4]);
> H := sub< G1 | phi(M) >;
> #H;
8