Paper 3, Section II, D

Groups | Part IA, 2011

(a) Let

SL2(Z)={(abcd)adbc=1,a,b,c,dZ}S L_{2}(\mathbb{Z})=\left\{\left(\begin{array}{ll} a & b \\ c & d \end{array}\right) \mid a d-b c=1, \quad a, b, c, d \in \mathbb{Z}\right\}

and, for a prime pp, let

SL2(Fp)={(abcd)adbc=1,a,b,c,dFp}S L_{2}\left(\mathbb{F}_{p}\right)=\left\{\left(\begin{array}{ll} a & b \\ c & d \end{array}\right) \mid a d-b c=1, \quad a, b, c, d \in \mathbb{F}_{p}\right\}

where Fp\mathbb{F}_{p} consists of the elements 0,1,,p10,1, \ldots, p-1, with addition and multiplication mod pp.

Show that SL2(Z)S L_{2}(\mathbb{Z}) and SL2(Fp)S L_{2}\left(\mathbb{F}_{p}\right) are groups under matrix multiplication.

[You may assume that matrix multiplication is associative, and that the determinant of a product equals the product of the determinants.]

By defining a suitable homomorphism from SL2(Z)SL2(F5)S L_{2}(\mathbb{Z}) \rightarrow S L_{2}\left(\mathbb{F}_{5}\right), show that

{(1+5a5b5c1+5d)SL2(Z)a,b,c,dZ}\left\{\left(\begin{array}{cc} 1+5 a & 5 b \\ 5 c & 1+5 d \end{array}\right) \in S L_{2}(\mathbb{Z}) \mid a, b, c, d \in \mathbb{Z}\right\}

is a normal subgroup of SL2(Z)S L_{2}(\mathbb{Z}).

(b) Define the group GL2(F5)G L_{2}\left(\mathbb{F}_{5}\right), and show that it has order 480 . By defining a suitable homomorphism from GL2(F5)G L_{2}\left(\mathbb{F}_{5}\right) to another group, which should be specified, show that the order of SL2(F5)S L_{2}\left(\mathbb{F}_{5}\right) is 120 .

Find a subgroup of GL2(F5)G L_{2}\left(\mathbb{F}_{5}\right) of index 2 .

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