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:

    “The first moments of sleep are an image of death; a hazy torpor grips our thoughts and it becomes impossible for us to determine the exact instant when the “I,” under another form, continues the task of existence.”
    —Gérard De Nerval (1808–1855)

    “We should never stand upon ceremony with sincerity. We should never cheat and insult and banish one another by our meanness, if there were present the kernel of worth and friendliness. We should not meet thus in haste.”
    —Henry David Thoreau (1817–1862)