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:
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:
“As every pool reflects the image of the sun, so every thought and thing restores us an image and creature of the supreme Good. The universe is perforated by a million channels for his activity. All things mount and mount.”
—Ralph Waldo Emerson (18031882)
“Thou shalt not, it is said, make unto thee any graven image of God. The same commandment should apply when God is taken to mean the living part of every human being, the part that cannot be grasped. It is a sin that, however much it is committed against us, we almost continually commit ourselvesExcept when we love.”
—Max Frisch (19111991)
“After nights thunder far away had rolled
The fiery day had a kernel sweet of cold”
—Edward Thomas (18781917)