The sign map sgn form S_n to {+1,-1} is a group morphism. Its kernel is the subgroup A_n of S_n.

The determinant is a morphism from the general linear group GL_n(R) to the multiplicative group R^*. Its kernel is the special linear group SL_n(R).

The map from Z to Z/nZ defined by x -> x mod n is a morphism of the additive group on Z onto Z/nZ. The kernel is nZ.

If H is a subgroup of a group G, then the map h H -> h G is a morphism. Its kernel is trivial, i.e., consists of the identity only.