General and special linear group/Determinant/Isomorphism theorem/Example

The determinant

$\det \colon \operatorname {GL} _{n}\!{\left(K\right)}\longrightarrow K^{\times },M\longmapsto \det M,$

is a surjective group homomorphism; its kernel is by definition the special linear group ${}\operatorname {SL} _{n}\!{\left(K\right)}$. Due to the isomorphism theorem, there exists a canonical isomorphism

${}\operatorname {GL} _{n}\!{\left(K\right)}/\operatorname {SL} _{n}\!{\left(K\right)}\cong K^{\times }\,.$