Special Linear Group $\text{SL}(m,\mathbb{Z}_n)$

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Definition (Special Linear Group $\text{SL}(m,\mathbb{Z}_n)$)

We have already defined the general linear group $\text{GL}(m,\mathbb{Z}_n)$ to be the set of $m$ by $m$ invertible matrices with coefficients in $\mathbb{Z}_n$, together with the binary operation of matrix multiplication mod $n$. The set of matrices in $\text{GL}(m,\mathbb{Z}_n)$ that have determinant 1 is called the special linear group, and we write $$\text{SL}(m,\mathbb{Z}_n)=\left\{ a\in \text{GL}(m,\mathbb{Z}_n) \mid \det(a)=1 \right\}.$$

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