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
- <math> h(u*v) = h(u) \cdot h(v) </math>
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,
- <math> h(e_G) = e_H</math>
and it also maps inverses to inverses in the sense that
- <math> h\left(u^{-1}\right) = h(u)^{-1}. \,</math>
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.
PropertiesEdit
Let <math>e_{H}</math> be the identity element of the (H, ·) group and <math>u \in G</math>, then
- <math>h(u) \cdot e_{H} = h(u) = h(u*e_{G}) = h(u) \cdot h(e_{G})</math>
Now by multiplying for the inverse of <math>h(u)</math> (or applying the cancellation rule) we obtain
- <math>e_{H} = h(e_{G})</math>
Similarly,
- <math> e_H = h(e_G) = h(u*u^{-1}) = h(u)\cdot h(u^{-1})</math>
Therefore for the uniqueness of the inverse: <math>h(u^{-1}) = h(u)^{-1}</math>.
TypesEdit
- MonomorphismTemplate:Anchor
- 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 kernelEdit
Template:Main article We define the kernel of h to be the set of elements in G which are mapped to the identity in H
- <math> \operatorname{ker}(h) := \left\{u \in G\colon h(u) = e_{H}\right\}.</math>
and the image of h to be
- <math> \operatorname{im}(h) := h(G) \equiv \left\{h(u)\colon u \in G\right\}.</math>
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 <math>u \in \operatorname{ker}(h)</math> and show <math>g^{-1} \circ u \circ g \in \operatorname{ker}(h)</math> for arbitrary <math>u, g</math>:
- <math>\begin{align}
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{align}</math> 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 Template:Nowrap}. Injection directly gives that there is a unique element in the kernel, and, conversely, a unique element in the kernel gives injection:
- <math>\begin{align}
&& 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{align}</math>
ExamplesEdit
- Consider the cyclic group Z3 = (Z/3Z, +) = ({0, 1, 2}, +) and the group of integers (Z, +). The map h : Z → Z/3Z with h(u) = u mod 3 is a group homomorphism. It is surjective and its kernel consists of all integers which are divisible by 3.
- The exponential map yields a group homomorphism from the group of real numbers R with addition to the group of non-zero real numbers R* with multiplication. The kernel is {0} and the image consists of the positive real numbers.
- The exponential map also yields a group homomorphism from the group of complex numbers C with addition to the group of non-zero complex numbers C* with multiplication. This map is surjective and has the kernel {2πki : k ∈ Z}, as can be seen from Euler's formula. Fields like R and C that have homomorphisms from their additive group to their multiplicative group are thus called exponential fields.
- The function <math>\Phi: (\mathbb{Z}, +) \rightarrow (\mathbb{R}, +)</math>, defined by <math>\Phi(x) = \sqrt[]{2}x</math> is a homomorphism.
- Consider the two groups <math>(\mathbb{R}^+, *)</math> and <math>(\mathbb{R}, +)</math>, represented respectively by <math>G</math> and <math>H</math>, where <math>\mathbb{R}^+</math> is the positive real numbers. Then, the function <math>f: G \rightarrow H </math> defined by the logarithm function is a homomorphism.
Category of groupsEdit
If Template:Nowrap and Template:Nowrap are group homomorphisms, then so is Template:Nowrap. 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 groupsEdit
If G and H are abelian (i.e., commutative) groups, then the set Template:Nowrap of all group homomorphisms from G to H is itself an abelian group: the sum Template:Nowrap 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 Template:Nowrap is again a group homomorphism.
The addition of homomorphisms is compatible with the composition of homomorphisms in the following sense: if f is in Template:Nowrap, h, k are elements of Template:Nowrap, and g is in Template:Nowrap, then
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 alsoEdit
ReferencesEdit
External linksEdit
- {{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web
|_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:GroupHomomorphism%7CGroupHomomorphism.html}} |title = Group Homomorphism |author = Rowland, Todd |website = MathWorld |access-date = |ref = Template:SfnRef }}