Paper 3, Section II, 7E7 \mathrm{E}

Groups | Part IA, 2012

Let Fp\mathbb{F}_{p} be the set of (residue classes of) integers modp\bmod p, and let

G={(abcd):a,b,c,dFp,adbc0}G=\left\{\left(\begin{array}{ll} a & b \\ c & d \end{array}\right): a, b, c, d \in \mathbb{F}_{p}, a d-b c \neq 0\right\}

Show that GG is a group under multiplication. [You may assume throughout this question that multiplication of matrices is associative.]

Let XX be the set of 2-dimensional column vectors with entries in Fp\mathbb{F}_{p}. Show that the mapping G×XXG \times X \rightarrow X given by

((abcd),(xy))(ax+bycx+dy)\left(\left(\begin{array}{ll} a & b \\ c & d \end{array}\right),\left(\begin{array}{l} x \\ y \end{array}\right)\right) \mapsto\left(\begin{array}{l} a x+b y \\ c x+d y \end{array}\right)

is a group action.

Let gGg \in G be an element of order pp. Use the orbit-stabilizer theorem to show that there exist x,yFpx, y \in \mathbb{F}_{p}, not both zero, with

g(xy)=(xy)g\left(\begin{array}{l} x \\ y \end{array}\right)=\left(\begin{array}{l} x \\ y \end{array}\right)

Deduce that gg is conjugate in GG to the matrix

(1101)\left(\begin{array}{ll} 1 & 1 \\ 0 & 1 \end{array}\right)

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