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)
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(eG)=eH
and it also maps inverses to inverses in the sense that
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 which 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.
The purpose of defining a group homomorphism is to create functions that preserve the algebraic structure. An equivalent definition of group homomorphism is: The function h : G → H is a group homomorphism if whenever
a ∗ b = c we have h(a) ⋅ h(b) = h(c).
In other words, the group H in some sense has a similar algebraic structure as G and the homomorphism h preserves that.
See main article: article, Image (mathematics) and kernel (algebra). We define the kernel of h to be the set of elements in G which are mapped to the identity in H
\operatorname{ker}(h):=\left\{u\inG\colonh(u)=eH\right\}.
and the image of h to be
\operatorname{im}(h):=h(G)\equiv\left\{h(u)\colonu\inG\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:
\begin{align} h\left(g-1\circu\circg\right)&=h(g)-1 ⋅ h(u) ⋅ h(g)\\ &=h(g)-1 ⋅ eH ⋅ h(g)\\ &=h(g)-1 ⋅ h(g)=eH, \end{align}
The homomorphism, h, is a group monomorphism; i.e., h is injective (one-to-one) if and only if . Injection directly gives that there is a unique element in the kernel, and, conversely, a unique element in the kernel gives injection:
\begin{align} &&h(g1)&=h(g2)\\ \Leftrightarrow&&h(g1) ⋅
| -1 | |
| h(g | |
| 2) |
&=eH\\ \Leftrightarrow&&h\left(g1\circ
| -1 | |
| g | |
| 2 |
\right)&=eH, \operatorname{ker}(h)=\{eG\}\\ ⇒ &&g1\circ
| -1 | |
| g | |
| 2 |
&=eG\\ \Leftrightarrow&&g1&=g2\end{align}
\Phi:(N,+) → (R,+)
\Phi(x)=\sqrt[]{2}x
(R+,*)
(R,+)
G
H
R+
f:G → H
If and are group homomorphisms, then so is . This shows that the class of all groups, together with group homomorphisms as morphisms, forms a category.
If G and H are abelian (i.e., commutative) groups, then the set of all group homomorphisms from G to H is itself an abelian group: the sum 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 is again a group homomorphism.
The addition of homomorphisms is compatible with the composition of homomorphisms in the following sense: if f is in, h, k are elements of, and g is in, then
and .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.