In mathematics, given two groups, (G,∗) and (H, ·), a group homomorphism from (G,∗) to (H, ·) is a function h : G → H such that for all u and v in G it holds that
h ( u ∗ v ) = h ( u ) ⋅ h ( v ) {\displaystyle h(u*v)=h(u)\cdot h(v)}
where the group operation on the left side of the equation is that of G and on the right side that of H. From this property, one can deduce that h maps the identity element eG of G to the identity element eH of H,
h ( e G ) = e H {\displaystyle h(e_{G})=e_{H}}
and it also maps inverses to inverses in the sense that
h ( u − 1 ) = h ( u ) − 1 . {\displaystyle h\left(u^{-1}\right)=h(u)^{-1}.\,}
Hence one can say that h "is compatible with the group structure". In areas of mathematics where one considers groups endowed with additional structure, a homomorphism sometimes means a map that respects not only the group structure (as above) but also the extra structure. For example, a homomorphism of topological groups is often required to be continuous.
Properties Let e H {\displaystyle e_{H}} be the identity element of the group (H, ·) and u ∈ G {\displaystyle u\in G} , then
h ( u ) ⋅ e H = h ( u ) = h ( u ∗ e G ) = h ( u ) ⋅ h ( e G ) {\displaystyle h(u)\cdot e_{H}=h(u)=h(u*e_{G})=h(u)\cdot h(e_{G})}
Now by multiplying by the inverse of h ( u ) {\displaystyle h(u)} (or applying the cancellation rule) we obtain
e H = h ( e G ) {\displaystyle e_{H}=h(e_{G})}
Similarly,
e H = h ( e G ) = h ( u ∗ u − 1 ) = h ( u ) ⋅ h ( u − 1 ) {\displaystyle e_{H}=h(e_{G})=h(u*u^{-1})=h(u)\cdot h(u^{-1})}
Therefore, by the uniqueness of the inverse: h ( u − 1 ) = h ( u ) − 1 {\displaystyle h(u^{-1})=h(u)^{-1}} .
Types Monomorphism A group homomorphism that is injective (or, one-to-one); i.e., preserves distinctness. Epimorphism A group homomorphism that is surjective (or, onto); i.e., reaches every point in the codomain. Isomorphism A group homomorphism that is bijective; i.e., injective and surjective. Its inverse is also a group homomorphism. In this case, the groups G and H are called isomorphic; they differ only in the notation of their elements (except of identity element) and are identical for all practical purposes. I.e. we re-label all elements except identity. Endomorphism A group homomorphism, h: G → G; the domain and codomain are the same. Also called an endomorphism of G. Automorphism A group endomorphism that is bijective, and hence an isomorphism. The set of all automorphisms of a group G, with functional composition as operation, itself forms a group, the automorphism group of G. It is denoted by Aut(G). As an example, the automorphism group of (Z, +) contains only two elements, the identity transformation and multiplication with −1; it is isomorphic to (Z/2Z, +).
Image and kernel
We define the kernel of h to be the set of elements in G that are mapped to the identity in H
ker ( h ) := { u ∈ G : h ( u ) = e H } . {\displaystyle \operatorname {ker} (h):=\left\{u\in G\colon h(u)=e_{H}\right\}.}
and the image of h to be
im ( h ) := h ( G ) ≡ { h ( u ) : u ∈ G } . {\displaystyle \operatorname {im} (h):=h(G)\equiv \left\{h(u)\colon u\in G\right\}.}
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