Skip to main content
\(\newcommand{\identity}{\mathrm{id}} \newcommand{\notdivide}{\nmid} \newcommand{\notsubset}{\not\subset} \newcommand{\lcm}{\operatorname{lcm}} \newcommand{\gf}{\operatorname{GF}} \newcommand{\inn}{\operatorname{Inn}} \newcommand{\aut}{\operatorname{Aut}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\cis}{\operatorname{cis}} \newcommand{\chr}{\operatorname{char}} \newcommand{\Null}{\operatorname{Null}} \newcommand{\transpose}{\text{t}} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \setcounter{chapter}{-1}\)



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?


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


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 \(D_n\text{?}\)


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 \(n / \gcd(k,n)\text{.}\)