6.6: Exercises

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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.)