1

Let $\sigma=(134)\text{,}$ $\tau=(23)(145)\text{,}$ $\rho=(56)(78)\text{,}$ and $\alpha=(12)(145)$ in $S_8\text{.}$ Compute the following.

1. $\sigma \tau$

2. $\tau \sigma$

3. $\tau^2$

4. $\tau^{-1}$

5. $o(\tau)$

6. $o(\rho)$

7. $o(\alpha)$

8. $\langle \tau\rangle$

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 $m$).

1. Prove that $H\leq S_n\text{.}$

2. 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)

1. have order 2?

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.)