Special linear group

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In mathematics, the special linear group ${\displaystyle \operatorname {SL} (n,R)}$ of degree ${\displaystyle n}$ over a commutative ring ${\displaystyle R}$ is the set of ${\displaystyle n\times n}$ matrices with determinant ${\displaystyle 1}$, with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the general linear group given by the kernel of the determinant

${\displaystyle \det \colon \operatorname {GL} (n,R)\to R^{\times }.}$

where ${\displaystyle R^{\times }}$ is the multiplicative group of ${\displaystyle R}$ (that is, ${\displaystyle R}$ excluding ${\displaystyle 0}$ when ${\displaystyle R}$ is a field).

These elements are "special" in that they form an algebraic subvariety of the general linear group – they satisfy a polynomial equation (since the determinant is polynomial in the entries).

When ${\displaystyle R}$ is the finite field of order ${\displaystyle q}$, the notation ${\displaystyle \operatorname {SL} (n,q)}$ is sometimes used.