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## Section1.3Exercises

###### 1

If the diagonals of a cube are labeled as Figure 5.26, to which motion of the cube does the permutation $(12)(34)$ correspond? What about the other permutations of the diagonals?

###### 2

Prove that $D_n$ is nonabelian for $n \geq 3\text{.}$

###### 3

Recall that the center of a group $G$ is

\begin{equation*} Z(G) = \{ g \in G : gx = xg \text{ for all } x \in G \}. \end{equation*}

Find the center of $D_8\text{.}$ What about the center of $D_{10}\text{?}$ What is the center of $D_n\text{?}$

###### 4

Let $r$ and $s$ be the elements in $D_n$ described in Theorem 5.23

1. Show that $srs = r^{-1}\text{.}$

2. Show that $r^k s = s r^{-k}$ in $D_n\text{.}$

3. Prove that the order of $r^k \in D_n$ is $n / \gcd(k,n)\text{.}$