Paper 3, Section II, D

(a) Let

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

and, for a prime $p$, let

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

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

Show that $S L_{2}(\mathbb{Z})$ and $S L_{2}\left(\mathbb{F}_{p}\right)$ 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 $S L_{2}(\mathbb{Z}) \rightarrow S L_{2}\left(\mathbb{F}_{5}\right)$, show that

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

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

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

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

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