Group Homomorphisms

Let \(G\) be a group and \(g \in G\). Define a map \(\phi : {\mathbb Z} \rightarrow G\) by \(\phi( n ) = g^n\). Then \(\phi\) is a group homomorphism, since \[\begin{aligned}\end{aligned}\]. This homomorphism maps \({\mathbb Z}\) onto the cyclic subgroup of \(G\) generated by \(g\).

Let \(G = GL_2( {\mathbb R })\). If \[\begin{aligned}\end{aligned}\] is in \(G\), then the determinant is nonzero; that is, \(\det(A) = ad - bc \neq 0\). Also, for any two elements \(A\) and \(B\) in \(G\), \(\det(AB) = \det(A) \det(B)\). Using the determinant, we can define a homomorphism \(\phi : GL_2( {\mathbb R }) \rightarrow {\mathbb R}^\ast\) by \(A \mapsto \det(A)\).

Recall that the circle group \({ \mathbb T}\) consists of all complex numbers \(z\) such that \(|z|=1\). We can define a homomorphism \(\phi\) from the additive group of real numbers \({\mathbb R}\) to \({\mathbb T}\) by \(\phi : \theta \mapsto \cos \theta + i \sin \theta\). Indeed, \[\begin{aligned}\phi( \alpha + \beta ) & = \cos( \alpha + \beta ) + i \sin( \alpha + \beta ) \\ & = (\cos \alpha \cos \beta - \sin \alpha \sin \beta) + i( \sin \alpha \cos \beta + \cos \alpha \sin \beta ) \\ & = (\cos \alpha + i \sin \alpha )(\cos \beta + i \sin \beta) \\ & = \phi( \alpha ) \phi( \beta )\end{aligned}\]. Geometrically, we are simply wrapping the real line around the circle in a group-theoretic fashion.