Editing Group homomorphism (section)

== Examples ==
* Consider the [[cyclic group]] Z{{sub|3}} = ('''Z'''/3'''Z''', +) = ({0, 1, 2}, +) and the group of integers ('''Z''', +). The map ''h'' : '''Z''' → '''Z'''/3'''Z''' with ''h''(''u'') = ''u'' [[modular arithmetic|mod]] 3 is a group homomorphism. It is [[surjective]] and its kernel consists of all integers which are divisible by 3.
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The set
:<math>G \equiv \left\{\begin{pmatrix}
    a & b \\
    0 & 1
  \end{pmatrix} \bigg| a > 0, b \in \mathbf{R}\right\}
</math>
forms a group under matrix multiplication. For any complex number ''u'' the function ''f<sub>u</sub>'' : ''G'' → '''C<sup>*</sup>''' defined by
:<math>\begin{pmatrix}
    a & b \\
    0 & 1
  \end{pmatrix} \mapsto a^u
</math>
is a group homomorphism.
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Consider a multiplicative group of [[positive real numbers]] ('''R'''<sup>+</sup>, ⋅) for any complex number ''u''. Then the function ''f<sub>u</sub>'' : '''R'''<sup>+</sup> → '''C''' defined by
:<math>f_u(a) = a^u</math>
is a group homomorphism.
}}
* The [[exponential function|exponential map]] yields a group homomorphism from the group of [[real number]]s '''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 number]]s '''C''' with addition to the group of non-zero complex numbers '''C'''* with multiplication. This map is surjective and has the kernel {2π''ki'' : ''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 field]]s.
* 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|logarithm function]] is a homomorphism.