split maximal torus construction

Question

In PGL(n,q) there is a split maximal torus T of order $(q-1)^{n-1}$. How to construct this in Magma?

Let's use the example of $PGL(4,11)$. I took a detour to construct it:

> G:=PGL(4,11);
> d:=DirectProduct(CyclicGroup(10),CyclicGroup(10));
> d:=DirectProduct(d,CyclicGroup(10));
> s:=Subgroups(G:OrderEqual:=10^3);
> S:=[s[i]`subgroup: i in [1..29]];
> [IsIsomorphic(d,S[i]): i in [1..29]];

And there is only one true isomorphism. So I got the $T$.

Is there a direct command or faster way to do this? I did some searching and didn't find any... Thank you.

Answer 1

The subgroup $T$ is the image of the subgroup of diagonal matrices in $GL(n,q)$.

MAGMA constructs $PGL(n,q)$ as a permutation group, acting on the projective line which has $(q^n - 1)/(q-1)$ points.

So in $PGL(n,q)$, think of $T$ as the stabilizer of $n$ points which correspond to a basis of $\mathbb{F}_q^n$.

In your example:

G := PGL(4,11);
T := Stabilizer(G, [1,2,3,4]);

should work.

I am not sure in general if in MAGMA the first $n$ points in the action of $PGL(n,q)$ correspond to a basis. I guess that would make sense, but this kind of information is not documented.

However you can verify to what vectors the points correspond to as follows:

G, I := PGL(4,11);
I[1..4];