Special linear group

Algebraic structure → Group theory Group theory
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Lie groups and Lie algebras
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Scientists Sophus Lie Henri Poincaré Wilhelm Killing Élie Cartan Hermann Weyl Claude Chevalley Harish-Chandra Armand Borel
Glossary Table of Lie groups
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In mathematics, the special linear group SL(n, R) of degree n over a commutative ring R is the set of n × n matrices with determinant 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 R^× is the multiplicative group of R (that is, R excluding 0 when 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 R is the finite field of order q, the notation SL(n, q) is sometimes used.