Computing preimages

Let $G=SL_2(5)$ and $p\colon G\to G/Z(G)$ be the canonical map. Typically, Magma will represent $G/Z(G)$ as a permutation group:

> G := SL(2,5);
> Z := Center(G);
> Q, p := quo< G|Z >;
> GroupName(Q);
A5

Let us compute some preimages. Let us take a random element y of Q, namely

> y := Random(Q);
> y;
(1, 3, 5, 4, 2)
> x := y @@ p;
> x;
[1 0]
[4 1]

How do I get all the preimages of an element $y$? One needs the (right) coset $(\ker p)x$, where $x$ is such that $p(x)=y$. For this, I need to convert my matrix group into a permutation group:

> f, P := PermutationRepresentation(G);

The map f we get is a bijective group homomorphism $SL_2(5)\to P$, where $P$ is a permutation group. The preimage of $y$ has size two, and it is the right coset $(\ker p)x$. Let us compute this:

> N := f(K);
> K := Kernel(p);

Now the preimage is the set

> { Inverse(f)(n*f(x)) : n in N };
{
    [1 0]
    [4 1],

    [4 0]
    [1 4]
}