Question

List two distinct cosets of < r > in , D₁₀ where r is a reflection in . D1

Answer 1

Taking into account that $r$ is a reflection, we conclude that $r^2=e,$ where $e$ is the identity of the dihedral group $D_{10},$ and hence $\langle r\rangle=\{e,r\}.$ Let $s$ be a 72 degree rotation counterclockwise in $D_{10}$. Then two distinct cosets of $\langle r\rangle$ are $e\langle r\rangle=\{e,r\}$ and $s\langle r\rangle=s\{e,r\}=\{s,sr\}.$