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

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    —Charles Sanders Peirce (1839–1914)

    “After night’s thunder far away had rolled
    The fiery day had a kernel sweet of cold”
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