Paper 3, Section II, D

Groups | Part IA, 2013

(a) Let pp be a prime, and let G=SL2(p)G=S L_{2}(p) be the group of 2×22 \times 2 matrices of determinant 1 with entries in the field Fp\mathbb{F}_{p} of integers modp\bmod p.

(i) Define the action of GG on X=Fp{}X=\mathbb{F}_{p} \cup\{\infty\} by Möbius 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 SL2(p)S L_{2}(p).

(ii) Let

A=(1101),B=(1301)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 AA is conjugate to BB in GG if p=11p=11, but not if p=5p=5.

(b) Let GG be the set of all 3×33 \times 3 matrices of the form

(1ax01b001)\left(\begin{array}{lll} 1 & a & x \\ 0 & 1 & b \\ 0 & 0 & 1 \end{array}\right)

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

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

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

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