3.2 Examples of Homomorphisms

Example 3.11

Write \(\mathbb {R} = (\mathbb {R},+)\) for the group of real numbers with the binary operation of addition. Write \(\mathbb {R}_+ = (\mathbb {R}_+, *)\) for the group of positive real numbers with the binary operation of multiplication. For \(x\in \mathbb {R}\), \(\exp x = e^x\in \mathbb {R}_+\). So we have a map \(\exp :\mathbb {R}\to \mathbb {R}_+\). From Calculus, we know that

\begin{equation} \label{exphom} \exp (x+y) = \exp (x) * exp(y) \end{equation}

3.12

for all \(x,y\in \mathbb {R}\). Therefore, \(\exp :\mathbb {R}\to \mathbb {R}_+\) is a group homomorphism.

We also learn in calculus that \(\exp :\mathbb {R}\to \mathbb {R}_+\) is one-one and onto with inverse \(\log :\mathbb {R}_+\to \mathbb {R}\).

Example 3.13

Define a map \(r:\mathbb {R}\to \operatorname{\mathrm{GL}}_2(\mathbb {R})\) by setting

\[ r(\theta ) := \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}. \]

Then, giving \(\mathbb {R}\) and \(\operatorname{\mathrm{GL}}_2(\mathbb {R})\) the obvious group structures (addition and matrix multiplication), \(r\) is a group homomorphism. This follows from the angle addition formulas for \(\cos \) and \(\sin \).