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:
“If the devil doesnt exist and, therefore, man has created him, he has created him in his own image and likeness.”
—Feodor Dostoyevsky (18211881)
“For me, the child is a veritable image of becoming, of possibility, poised to reach towards what is not yet, towards a growing that cannot be predetermined or prescribed. I see her and I fill the space with others like her, risking, straining, wanting to find out, to ask their own questions, to experience a world that is shared.”
—Maxine Greene (20th century)
“After nights thunder far away had rolled
The fiery day had a kernel sweet of cold”
—Edward Thomas (18781917)