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Kernels and Images of Homomorphisms

Given two groupsanda group homomorphism fromtois a functionsuch that for all

where the group operation on the left hand side of the equation is that ofand on the right hand side that of

maps the identity elementof G to the identity elementofand 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 ofsinceand the image is a subgroup of H. The homomorphism h is injective if and only if

Examples

  • The cyclic groupand the group of integersunder addition. The mapwithis 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 isand the image consists of the positive real numbers.

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