Abstract Algebra: Theory and Applications · Problem 9.4.13

Normal Subgroups and Factor Groups — Thomas Judson

Mathematics — Abstract Algebra · dihedral-group-isomorphism

Problem

Let

\omega = \text{cis}(2\pi/n)

be a primitive

n

th root of unity. Prove that the matrices

A = \begin{pmatrix} \omega & 0 \\ 0 & \omega^{-1} \end{pmatrix} \quad \text{and} \quad B = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}

generate a multiplicative group isomorphic to

D_n