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, image and/or kernel:

    We have no participation in Being, because all human nature is ever midway between being born and dying, giving off only a vague image and shadow of itself, and a weak and uncertain opinion. And if you chance to fix your thoughts on trying to grasp its essence, it would be neither more nor less than if your tried to clutch water.
    Michel de Montaigne (1533–1592)

    O comfort-killing night, image of hell,
    Dim register and notary of shame,
    Black stage for tragedies and murders fell,
    Vast sin-concealing chaos, nurse of blame!
    William Shakespeare (1564–1616)

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