6.6: Exercises
Throughout, write all permutations using disjoint cycle notation , and write all dihedral group elements in standard form .
- Let \(\sigma=(134)\text{,}\) \(\tau=(23)(145)\text{,}\) \(\rho=(56)(78)\text{,}\) and \(\alpha=(12)(145)\) in \(S_8\text{.}\) Compute the following.
- \(\sigma \tau\)
- \(\tau \sigma\)
- \(\tau^2\)
- \(\tau^{-1}\)
- \(o(\tau)\)
- \(o(\rho)\)
- \(o(\alpha)\)
- \(\langle \tau\rangle\)
2. Prove Lemma 6.3.4.
3. Prove that \(A_n\) is a subgroup of \(S_n\text{.}\)
4. Prove or disprove: The set of all odd permutations in \(S_n\) is a subgroup of \(S_n\text{.}\)
5. Let \(n\) be an integer greater than 2. \(m \in \{1,2,\ldots,n\}\text{,}\) and let \(H=\{\sigma\in S_n\,:\,\sigma(m)=m\}\) (in other words, \(H\) is the set of all permutations in \(S_n\) that fix \(m\)).
- Prove that \(H\leq S_n\text{.}\)
- Identify a familiar group to which \(H\) is isomorphic. (You do not need to show any work.)
6. Write \(rfr^2frfr\) in \(D_5\) in standard form.
7. Prove or disprove: \(D_6\simeq S_6\text{.}\)
8. Which elements of \(D_4\) (if any)
- have order \(2\)?
- have order \(3\text{?}\)
9. Let \(n\) be an even integer that's greater than or equal to \(4\). Prove that \(r^{n/2}\in Z(D_n)\text{:}\) that is, prove that \(r^{n/2}\) commutes with every element of \(D_n\text{.}\) (Do NOT simply refer to the last statement in Theorem \(6.5.4\); that is the statement you are proving here.)