Group homomorphism

Homomorphisme de groupes (Fr). Gruppenhomomorphismus (Ge). Homomorfismo de grupos (Sp). Omomorfismo di gruppi (It). 準同形 (Ja).

Groups

Let G and H be two non-empty sets with binary operations * (in G) and # (in H). If * and # are associative in G and H respectively and if G and H contain an identity element and the inverse of each element in them, then (G, *) and (H, #) are two groups.

Homomorphism between groups

A group homomorphism from (G, *) to (H, #) is a function f : G → H that preserves the composition law, i.e. such that for all u and v in G:

f(u * v) = f(u) # f(v).

The function f maps the identity element 1_G of G to the identity element 1_H of H, and it also maps inverses to inverses: f(u^-1) = f(u)^-1.

Kernel and image

Group homomorphism. The kernel of the homomorphism, ker( f ) is the set of elements of G that are mapped on the identity element of H. The image of the homomorphism, im( f ), is the set of elements of H with one or more corresponding element in G. im( f ) is not required to span the whole H

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The kernel of the homomorphism is the set of the elements of G that are mapped on the identity of H:

ker( f ) = { u in G : f(u) = 1_H }

The image of the homomorphism is the subset of elements of H that are mapped by the homomorphism f:

im( f ) = { f(u) : u in G }.

The kernel is a normal subgroup and the image is a subgroup of H.

Types of homomorphisms

Homomorphisms can be classified according to different criteria, among which the relation between G and H and the nature of the mapping.

Homomorphisms mapping elements of different sets (G ≠ H)

An epimorphism is a surjective homomorphism, i.e. a homomorphism where the image spans the whole codomain

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An epimorphism is a surjective homomorphism, that is, a homomorphism which is onto as a function. The image of the homomorphism spans the whole set H: in this case, img( f ) = H

A monomorphism is an injective homomorphism, i.e. a homomorphism where the the elements of H are mapped to one and only one element of G

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A monomorphism is an injective homomorphism, that is, a homomorphism which is one-to-one as a function. In this case, ker( f ) = {1_G }.

An isomorphism is a bijective homomorphism, i.e. there is a one-to-one correspondence between the elements of G and those of H. Groups (G,*) and (H,#) differ only for the notation of their elements

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If the homomorphism f is a bijection, then its inverse is also a group homomorphism, and f 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.

Homomorphisms mapping elements of the same set (G = H)

An endomorphism is homomorphism of G to itself

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An endomorphism is a homomorphism of a group onto itself: f: G → G.

A bijective (invertible) endomorphism (which is hence an isomorphism) is called an automorphism. The kernel of the automorphism is the identity of G (1_G) and the image of the automorphism coincides with G. The set of all automorphisms of a group (G,*) forms itself a group, the automorphism group of G, Aut(G).