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## Perform computations in a quotient of dihedral group of order 16

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.17

Let $G = D_{16}$ and let $H = \langle r^4 \rangle$.

(1) Show that the order of $G/H$ is 8.
(2) Exhibit each element of $G/H$ as $\overline{s}^a \overline{r}^b$ for some integers $a$ and $b$.
(3) Find the order of each of the elements exhibited in the previous point.
(4) Write each of the following elements as $\overline{s}^a \overline{r}^b$: $\overline{rs}$, $\overline{sr^{-2}s}$, $\overline{s^{-1}r^{-1}sr}$.
(5) Prove that $\overline{K} = \langle \overline{s}, \overline{r}^2 \rangle$ is a normal subgroup of $G/H$ and that $\overline{K}$ is isomorphic to $V_4$. Describe the isomorphism type of $K = \langle s, r^2 \rangle$ in $G$.
(6) Find the center of $G/H$ and describe the isomorphism type of $(G/H)/Z(G/H)$.