Abstract Algebra: Theory and Applications · Problem 9.4.2

Normal Subgroups and Factor Groups — Thomas Judson

Mathematics — Abstract Algebra · isomorphism-proof

Problem

Prove that

\mathbb{C}^*

is isomorphic to the subgroup of

GL_2(\mathbb{R})

consisting of matrices of the form

\begin{pmatrix} a & b \\ -b & a \end{pmatrix}.

Hint. Define

\phi : \mathbb{C}^* \to GL_2(\mathbb{R})

\phi(a + bi) = \begin{pmatrix} a & b \\ -b & a \end{pmatrix}.