Group Homomorphism - Examples

Examples

  • Consider the cyclic group Z/3Z = {0, 1, 2} and the group of integers Z with addition. 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.
  • Consider and group G:=\left\{\begin{pmatrix} a & b \\ 0 & 1 \end{pmatrix}\mid a>0,b\in\mathbb{R}\right\} with . then functions of the form
s.t. f_u \left(\begin{pmatrix} a & b \\ 0 & 1 \end{pmatrix}\right)=a^u are group homomorphisms.
  • Consider multiplicative group of positive real numbers with then functions of the form
s.t. are group homomorphisms.
  • 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 in 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.

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