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)} {\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}} {\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}.\,} {\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}} {\displaystyle e_{H}} be the identity element of the group (H, ·) and u ∈ G {\displaystyle u\in 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})} {\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)} {\displaystyle h(u)} (or applying the cancellation rule) we obtain

e H = h ( e G ) {\displaystyle 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})} {\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}} {\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\}.} {\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\}.} {\displaystyle \operatorname {im} (h):=h(G)\equiv \left\{h(u)\colon u\in G\right\}.}

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.

The kernel of h is a normal subgroup of G. Assume u ∈ ker ⁡ ( h ) {\displaystyle u\in \operatorname {ker} (h)} {\displaystyle u\in \operatorname {ker} (h)} and show g − 1 ∘ u ∘ g ∈ ker ⁡ ( h ) {\displaystyle g^{-1}\circ u\circ g\in \operatorname {ker} (h)} {\displaystyle g^{-1}\circ u\circ g\in \operatorname {ker} (h)} for arbitrary u , g {\displaystyle u,g} {\displaystyle u,g}:

h ( g − 1 ∘ u ∘ g ) = h ( g ) − 1 ⋅ h ( u ) ⋅ h ( g ) = h ( g ) − 1 ⋅ e H ⋅ h ( g ) = h ( g ) − 1 ⋅ h ( g ) = e H , {\displaystyle {\begin{aligned}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},\end{aligned}}} {\displaystyle {\begin{aligned}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},\end{aligned}}}

The image of h is a subgroup of H.

The homomorphism, h, is a group monomorphism; i.e., h is injective (one-to-one) if and only if ker(h) = {eG}. Injectivity directly gives that there is a unique element in the kernel, and, conversely, a unique element in the kernel gives injectivity:

h ( g 1 ) = h ( g 2 ) ⇔ h ( g 1 ) ⋅ h ( g 2 ) − 1 = e H ⇔ h ( g 1 ∘ g 2 − 1 ) = e H ,   ker ⁡ ( h ) = { e G } ⇒ g 1 ∘ g 2 − 1 = e G ⇔ g 1 = g 2 {\displaystyle {\begin{aligned}&&h(g_{1})&=h(g_{2})\\\Leftrightarrow &&h(g_{1})\cdot h(g_{2})^{-1}&=e_{H}\\\Leftrightarrow &&h\left(g_{1}\circ g_{2}^{-1}\right)&=e_{H},\ \operatorname {ker} (h)=\{e_{G}\}\\\Rightarrow &&g_{1}\circ g_{2}^{-1}&=e_{G}\\\Leftrightarrow &&g_{1}&=g_{2}\end{aligned}}} {\displaystyle {\begin{aligned}&&h(g_{1})&=h(g_{2})\\\Leftrightarrow &&h(g_{1})\cdot h(g_{2})^{-1}&=e_{H}\\\Leftrightarrow &&h\left(g_{1}\circ g_{2}^{-1}\right)&=e_{H},\ \operatorname {ker} (h)=\{e_{G}\}\\\Rightarrow &&g_{1}\circ g_{2}^{-1}&=e_{G}\\\Leftrightarrow &&g_{1}&=g_{2}\end{aligned}}}

Examples

Category of groups

If h : G → H and k : H → K are group homomorphisms, then so is k ∘ h : G → K. This shows that the class of all groups, together with group homomorphisms as morphisms, forms a category (specifically the category of groups).

Homomorphisms of abelian groups

If G and H are abelian (i.e., commutative) groups, then the set Hom(G, H) of all group homomorphisms from G to H is itself an abelian group: the sum h + k of two homomorphisms is defined by

(h + k)(u) = h(u) + k(u)    for all u in G.

The commutativity of H is needed to prove that h + k is again a group homomorphism.

The addition of homomorphisms is compatible with the composition of homomorphisms in the following sense: if f is in Hom(K, G), h, k are elements of Hom(G, H), and g is in Hom(H, L), then

(h + k) ∘ f = (h ∘ f) + (k ∘ f)    and    g ∘ (h + k) = (g ∘ h) + (g ∘ k).

Since the composition is associative, this shows that the set End(G) of all endomorphisms of an abelian group forms a ring, the endomorphism ring of G. For example, the endomorphism ring of the abelian group consisting of the direct sum of m copies of Z/nZ is isomorphic to the ring of m-by-m matrices with entries in Z/nZ. The above compatibility also shows that the category of all abelian groups with group homomorphisms forms a preadditive category; the existence of direct sums and well-behaved kernels makes this category the prototypical example of an abelian category.

See also

References