Example

Let G be the set of all 2×2 matrices with entries from a field F of the form

1 * 0 *

where * represents an arbitrary element of F.

Then G is a subgroup of GL(F²). The subgroup

N =

1 * 0 1

is a normal subgroup of G. The quotient group G/N is isomorphic to the multiplicative group on F\{0}.

Observe that N is the kernel of the determinant, viewed as a morphism G -> F\{0}.