Since Z is commutative, its subgroup nZ is normal. It is the kernel of the surjective morphism

Z -> Z/nZ,      z -> z+(n).

The quotient of this group is the group Z/nZ. The notation here fits with the notation introduced in Chapter 2.

Let G be a group and N a normal subgroup of G distinct from {e} and from G. The groups G/N and N are both smaller than G. A lot of information about G can be obtained from study of these two smaller groups. However, the exact structure of G is not completely determined by G/N and N. For instance, the groups Z/4Z and Z/2Z × Z/2Z both have a normal subgroup isomorphic with Z/2Z, and in both cases the quotient group is isomorphic with Z/2Z.

The group S_n/A_n is isomorphic with Z/2Z. For, the map sgn : S_n-> {-1,1} is a surjective morphism with kernel A_n. Here, {1,-1} is the group of invertible elements of the ring Z. This group is isomorphic with Z/2Z.

The group GL_n(R)/SL_n(R) is isomorphic with R^*. Here we find SL_n(R) as the kernel of the determinant map det : GL_n(R) -> R^*.