Group Homomorphism - Image and Kernel

Image and Kernel

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 of h is a normal subgroup of G and the image of h is a subgroup of H:

h\left(g^{-1} \circ u\circ g\right)= h(g)^{-1}\cdot h(u)\cdot h(g) = h(g)^{-1}\cdot e_H\cdot h(g) = h(g)^{-1}\cdot h(g) = e_H.

The homomorphism h is injective (and called a group monomorphism) if and only if ker(h) = {eG}.

The kernel and image of a homomorphism can be interpreted as measuring how close it is to being an isomorphism. The First Isomorphism Theorem states that the image of a group homomorphism, h(G) is isomorphic to the quotient group G/ker h.

Read more about this topic:  Group Homomorphism

Famous quotes containing the words image and/or kernel:

    “Nowadays people’s visual imagination is so much more sophisticated, so much more developed, particularly in young people, that now you can make an image which just slightly suggests something, they can make of it what they will.”
    —Robert Doisneau (b. 1912)

    “After night’s thunder far away had rolled
    The fiery day had a kernel sweet of cold”
    —Edward Thomas (1878–1917)