Group homomorphism
From Online Dictionary of Crystallography
Homomorphisme de groupes (Fr). Gruppenhomomorphismus (Ge). Homomorfismo de grupos (Sp). Omomorfismo di gruppi (It). 準同形 (Ja).
A group homomorphism from (G, *) to (H, ·) is a function h : G → H such that for all u and v in G:
h(u * v) = h(u) · h(v)
where the group operation on the left hand side of the equation is that of G and on the right hand side that of H.
The operation h maps the identity element 1G of G to the identity element 1H of H, and it also maps inverses to inverses: h(u-1) = h(u)-1.
Image and kernel
The kernel of h is defined as:
ker(h) = { u in G : h(u) = 1H }
The image of h is defined as:
im(h) = { h(u) : u in G }.
The kernel is a normal subgroup and the image is a subgroup of H.
Types of homomorphisms
- If the homomorphism h is a bijection, then its inverse is also a group homomorphism, and h is called an isomorphism; the groups G and H are called isomorphic and differ only in the notation of their elements, while they are identical for all practical purposes.
- An endomorphism is a homomorphism of a group onto itself: h: G → G.
- A bijective (invertible) endomorphism (which is hence an isomorphism) is called an automorphism. The set of all automorphisms of a group G forms itself a group, the automorphism group of G, Aut(G).
- An epimorphism is a surjective homomorphism, that is, a homomorphism which is onto as a function.
- A monomorphism is an injective homomorphism, that is, a homomorphism which is one-to-one as a function. In this case, ker(h) = {1G }.