## Kernels and Images of Homomorphisms

Given two groups and a group homomorphism from to is a function such that for all where the group operation on the left hand side of the equation is that of and on the right hand side that of  maps the identity element of G to the identity element of and it also maps inverses to inverses: We define the kernel of h to be the set of elements in G which are mapped to the identity in H and the image of h to be The kernel is a normal subgroup of since and the image is a subgroup of H. The homomorphism h is injective if and only if Examples

• The cyclic group and the group of integers under addition. The map with is a group homomorphism. It is surjective and the kernel consists of all multiples of 3.

• The exponential map yields a group homomorphism from the group of real numbers under addition to the group of non-zero real numbers - the image - under multiplication. The kernel is and the image consists of the positive real numbers.

• The exponential map also yields a group homomorphism from the group of complex numbers with addition to the group of non-zero complex numbers with multiplication. This map is surjective and has the kernel sine  