Paper 3, Section II, D

Groups | Part IA, 2014

Let pp be a prime number, and G=GL2(Fp)G=G L_{2}\left(\mathbb{F}_{p}\right), the group of 2×22 \times 2 invertible matrices with entries in the field Fp\mathbb{F}_{p} of integers modulo pp.

The group GG acts on X=Fp{}X=\mathbb{F}_{p} \cup\{\infty\} by Möbius transformations,

(abcd)z=az+bcz+d\left(\begin{array}{ll} a & b \\ c & d \end{array}\right) \cdot z=\frac{a z+b}{c z+d}

(i) Show that given any distinct x,y,zXx, y, z \in X there exists gGg \in G such that g0=xg \cdot 0=x, g1=yg \cdot 1=y and g=zg \cdot \infty=z. How many such gg are there?

(ii) GG acts on X×X×XX \times X \times X by g(x,y,z)=(gx,gy,gz)g \cdot(x, y, z)=(g \cdot x, g \cdot y, g \cdot z). Describe the orbits, and for each orbit, determine its stabiliser, and the orders of the orbit and stabiliser.

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