# Paper 3, Section II, D

(a) Let $p$ be a prime, and let $G=S L_{2}(p)$ be the group of $2 \times 2$ matrices of determinant 1 with entries in the field $\mathbb{F}_{p}$ of integers $\bmod p$.

(i) Define the action of $G$ on $X=\mathbb{F}_{p} \cup\{\infty\}$ by MÃ¶bius transformations. [You need not show that it is a group action.]

State the orbit-stabiliser theorem.

Determine the orbit of $\infty$ and the stabiliser of $\infty$. Hence compute the order of $S L_{2}(p)$.

(ii) Let

$A=\left(\begin{array}{ll} 1 & 1 \\ 0 & 1 \end{array}\right), \quad B=\left(\begin{array}{ll} 1 & 3 \\ 0 & 1 \end{array}\right)$

Show that $A$ is conjugate to $B$ in $G$ if $p=11$, but not if $p=5$.

(b) Let $G$ be the set of all $3 \times 3$ matrices of the form

$\left(\begin{array}{lll} 1 & a & x \\ 0 & 1 & b \\ 0 & 0 & 1 \end{array}\right)$

where $a, b, x \in \mathbb{R}$. Show that $G$ is a subgroup of the group of all invertible real matrices.

Let $H$ be the subset of $G$ given by matrices with $a=0$. Show that $H$ is a normal subgroup, and that the quotient group $G / H$ is isomorphic to $\mathbb{R}$.

Determine the centre $Z(G)$ of $G$, and identify the quotient group $G / Z(G)$.