A group homomorphism is a function between groups which "respects the group structure".
Definition
Formally, given two groups (G,+) and (H,∗) (which hereafter we will abbreviate as G and H respectively), a group homomorphism from G to H is a function f from the underlying set G to the underlying set H, such that f(a)∗f(b)=f(a+b) for all a,b∈G.
Examples
- For any group G, there is a group homomorphism 1G:G→G, given by 1G(g)=g for all g∈G. This homomorphism is always bijective.
- For any group G, there is a (unique) group homomorphism into the group {e} with one element and the only possible group operation e∗e=e. This homomorphism is given by g↦e for all g∈G. This homomorphism is usually not injective: it is injective if and only if G is the group with one element. (Uniqueness is guaranteed because there is only one function, let alone group homomorphism, from any set X to a set with one element.)
- For any group G, there is a (unique) group homomorphism from the group with one element into G, given by e↦eG, the identity of G. This homomorphism is usually not surjective: it is surjective if and only if G is the group with one element. (Uniqueness is guaranteed this time by the property proved below that the identity gets mapped to the identity.)
- For any group (G,+), there is a bijective group homomorphism to another group Gop given by taking inverses: g↦g−1. The group Gop is defined to have underlying set equal to that of G, and group operation g+oph:=h+g.
- For any pair of groups G,H, there is a homomorphism between G and H given by g↦eH.
- There is only one homomorphism between the group C2={eC2,g} with two elements and the group C3={eC3,h,h2} with three elements; it is given by eC2↦eC3,g↦eC3. For example, the function f:C2→C3 given by eC2↦eC3,g↦h is not a group homomorphism, because if it were, then eC3=f(eC2)=f(gg)=f(g)f(g)=hh=h2, which is not true. (We have used that the identity gets mapped to the identity.)
Properties
- The identity gets mapped to the identity. (Proof.)
- The inverse of the image is the image of the inverse. (Proof.)
- The image of a group under a homomorphism is another group. (Proof.)
- The composition of two homomorphisms is a homomorphism. (Proof.)