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/or kernel:
“on the instant clamorous eaves,
A climbing moon upon an empty sky,
And all that lamentation of the leaves,
Could but compose mans image and his cry.”
—William Butler Yeats (18651939)
“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 (18201849)