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Suppose $N=p_1^{n_1}p_2^{n_2}...p_t^{n_t} $ where the $p_i$ are unique primes .

Now by the Chinese Remainder Theorem

$$ SL_2(\mathbb{Z}/N\mathbb{Z})=SL_2(\mathbb{Z}/p_1^{n_1}\mathbb{Z}) \times SL_2(\mathbb{Z}/p_2^{n_2} \mathbb{Z}) \times...\times SL_2(\mathbb{Z}/p_t^{n_t}\mathbb{Z})$$

I only know that $$ \mathbb{Z}/N\mathbb{Z}=\mathbb{Z}/p_1^{n_1}\mathbb{Z} \times \mathbb{Z}/p_2^{n_2}\mathbb{Z} \times... \times \mathbb{Z}/p_t^{n_t}\mathbb{Z}$$ by the Chinese Remainder Theorem , but why is it also true for $SL_2(\mathbb{Z})=\lbrace \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix} :a,b,c,d \in \mathbb{Z} \ \ , ad-bc=1 \rbrace $ ?

Thanks for help .

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  • $\begingroup$ What is $SL_2$? $\endgroup$ Jun 2, 2019 at 11:03
  • $\begingroup$ $D\equiv 1\pmod{N}\iff(\forall i)(D\equiv 1\pmod{p_i^{n_i}})$. $\endgroup$
    – metamorphy
    Jun 2, 2019 at 11:20

1 Answer 1

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Because, more generally, for any commutative rings $R$ and $S$, we have an isomorphism of matrix rings $$\operatorname{M}_{n×n} (R × S) \cong \operatorname{M}_{n×n} R × \operatorname{M}_{n×n} S,$$ that restricts to isomorphisms of matrix groups $$\operatorname{GL}_n (R × S) \cong \operatorname{GL}_n R × \operatorname{GL}_n S \quad\text{and}\quad \operatorname{SL}_n (R × S) \cong \operatorname{SL}_n R × \operatorname{SL}_n S.$$ The first isomorphism, the one of the matrix rings, should be fairly obvious: Just split up a matrix of component pairs into a pair of component matrices. No information is lost, addition and multiplication are preserved. Why does it restrict so nicely to these matrix groups?

It’s because $(R×S)^× = R^× × S^×$ and $(1,1)$ is the unit element in $R × S$ and the determinant of a matrix $A ∈ \operatorname{M}_{n×n} (R×S)$ is given by the determinants of its component matrices $A_R$ and $A_S$, that is $$\det A = (\det A_R, \det A_S).$$

Hence, the isomorphism restricts to the general linear and then to the special linear group. You can remember this as “matrix functors respect products”.

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