The dihedral group $D_8$ isn't Hamiltonian

Question

Let $D_8=\{a^ib^j:i\in\{0,1\},j\in\{0,...,3\}\} $ be a dihedral group, where $$a=\begin{pmatrix} -1 &0 \\ 0 & 1 \end{pmatrix}\qquad\text{ and }\qquad b=\begin{pmatrix} cos\theta & -sin\theta \\ sin\theta & cos\theta \end{pmatrix},$$ with $\theta=\frac{2\pi}{4}$. Prove that $D_8$ is not a Hamiltonian group.

Answer 1

To show that $D_8$ is not Hamiltonian, it suffices to find a subgroup that is not normal. Since $D_8$ has only $8$ elements, it isn't a lot of work to check all cyclic subgroups, i.e. to check $\langle x\rangle$ for all $x\in D_8$.