Throughout, write all permutations using *disjoint cycle notation*, and write all dihedral group elements in *standard form*.

##### 1

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 6.5.10; that is the statement you are proving here.)