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:

    “He took the gold from them, formed it in a mold, and cast an image of a calf; and they said, “These are your gods, O Israel, who brought you up out of the land of Egypt!””
    —Bible: Hebrew, Exodus 32:4.

    “All true histories contain instruction; though, in some, the treasure may be hard to find, and when found, so trivial in quantity that the dry, shrivelled kernel scarcely compensates for the trouble of cracking the nut.”
    —Anne Brontë (1820–1849)