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## Exercises5.4Exercises

###### 1.

Write the following permutations in cycle notation.

1. \begin{equation*} \begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 2 & 4 & 1 & 5 & 3 \end{pmatrix} \end{equation*}
2. \begin{equation*} \begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 4 & 2 & 5 & 1 & 3 \end{pmatrix} \end{equation*}
3. \begin{equation*} \begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 3 & 5 & 1 & 4 & 2 \end{pmatrix} \end{equation*}
4. \begin{equation*} \begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 1 & 4 & 3 & 2 & 5 \end{pmatrix} \end{equation*}
###### 2.

Compute each of the following.

1. $\displaystyle (1345)(234)$

2. $\displaystyle (12)(1253)$

3. $\displaystyle (143)(23)(24)$

4. $\displaystyle (1423)(34)(56)(1324)$

5. $\displaystyle (1254)(13)(25)$

6. $\displaystyle (1254) (13)(25)^2$

7. $\displaystyle (1254)^{-1} (123)(45) (1254)$

8. $\displaystyle (1254)^2 (123)(45)$

9. $\displaystyle (123)(45) (1254)^{-2}$

10. $\displaystyle (1254)^{100}$

11. $\displaystyle |(1254)|$

12. $\displaystyle |(1254)^2|$

13. $\displaystyle (12)^{-1}$

14. $\displaystyle (12537)^{-1}$

15. $\displaystyle [(12)(34)(12)(47)]^{-1}$

16. $\displaystyle [(1235)(467)]^{-1}$

###### 3.

Express the following permutations as products of transpositions and identify them as even or odd.

1. $\displaystyle (14356)$

2. $\displaystyle (156)(234)$

3. $\displaystyle (1426)(142)$

4. $\displaystyle (17254)(1423)(154632)$

5. $\displaystyle (142637)$

###### 4.

Find $(a_1, a_2, \ldots, a_n)^{-1}\text{.}$

###### 5.

List all of the subgroups of $S_4\text{.}$ Find each of the following sets:

1. $\displaystyle \{ \sigma \in S_4 : \sigma(1) = 3 \}$

2. $\displaystyle \{ \sigma \in S_4 : \sigma(2) = 2 \}$

3. $\{ \sigma \in S_4 : \sigma(1) = 3$ and $\sigma(2) = 2 \}\text{.}$

Are any of these sets subgroups of $S_4\text{?}$

###### 6.

Find all of the subgroups in $A_4\text{.}$ What is the order of each subgroup?

###### 7.

Find all possible orders of elements in $S_7$ and $A_7\text{.}$

###### 8.

Show that $A_{10}$ contains an element of order $15\text{.}$

###### 9.

Does $A_8$ contain an element of order $26\text{?}$

###### 10.

Find an element of largest order in $S_n$ for $n = 3, \ldots, 10\text{.}$

###### 11.

What are the possible cycle structures of elements of $A_5\text{?}$ What about $A_6\text{?}$

###### 12.

Let $\sigma \in S_n$ have order $n\text{.}$ Show that for all integers $i$ and $j\text{,}$ $\sigma^i = \sigma^j$ if and only if $i \equiv j \pmod{n}\text{.}$

###### 13.

Let $\sigma = \sigma_1 \cdots \sigma_m \in S_n$ be the product of disjoint cycles. Prove that the order of $\sigma$ is the least common multiple of the lengths of the cycles $\sigma_1, \ldots, \sigma_m\text{.}$

###### 14.

Using cycle notation, list the elements in $D_5\text{.}$ What are $r$ and $s\text{?}$ Write every element as a product of $r$ and $s\text{.}$

###### 15.

If the diagonals of a cube are labeled as Figure 5.28, to which motion of the cube does the permutation $(12)(34)$ correspond? What about the other permutations of the diagonals?

###### 16.

Find the group of rigid motions of a tetrahedron. Show that this is the same group as $A_4\text{.}$

###### 17.

Prove that $S_n$ is nonabelian for $n \geq 3\text{.}$

###### 18.

Show that $A_n$ is nonabelian for $n \geq 4\text{.}$

###### 19.

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

###### 20.

Let $\sigma \in S_n$ be a cycle. Prove that $\sigma$ can be written as the product of at most $n-1$ transpositions.

###### 21.

Let $\sigma \in S_n\text{.}$ If $\sigma$ is not a cycle, prove that $\sigma$ can be written as the product of at most $n - 2$ transpositions.

###### 22.

If $\sigma$ can be expressed as an odd number of transpositions, show that any other product of transpositions equaling $\sigma$ must also be odd.

###### 23.

If $\sigma$ is a cycle of odd length, prove that $\sigma^2$ is also a cycle.

###### 24.

Show that a $3$-cycle is an even permutation.

###### 25.

Prove that in $A_n$ with $n \geq 3\text{,}$ any permutation is a product of cycles of length $3\text{.}$

###### 26.

Prove that any element in $S_n$ can be written as a finite product of the following permutations.

1. $\displaystyle (1 2), (13), \ldots, (1n)$

2. $\displaystyle (1 2), (23), \ldots, (n- 1,n)$

3. $\displaystyle (12), (1 2 \ldots n )$

###### 27.

Let $G$ be a group and define a map $\lambda_g : G \rightarrow G$ by $\lambda_g(a) = g a\text{.}$ Prove that $\lambda_g$ is a permutation of $G\text{.}$

###### 28.

Prove that there exist $n!$ permutations of a set containing $n$ elements.

###### 29.

Recall that the center of a group $G$ is

\begin{equation*} Z(G) = \{ g \in G : gx = xg \text{ for all } x \in G \}\text{.} \end{equation*}

Find the center of $D_8\text{.}$ What about the center of $D_{10}\text{?}$ What is the center of $D_n\text{?}$

###### 30.

Let $\tau = (a_1, a_2, \ldots, a_k)$ be a cycle of length $k\text{.}$

1. Prove that if $\sigma$ is any permutation, then

\begin{equation*} \sigma \tau \sigma^{-1 } = ( \sigma(a_1), \sigma(a_2), \ldots, \sigma(a_k)) \end{equation*}

is a cycle of length $k\text{.}$

2. Let $\mu$ be a cycle of length $k\text{.}$ Prove that there is a permutation $\sigma$ such that $\sigma \tau \sigma^{-1 } = \mu\text{.}$

###### 31.

For $\alpha$ and $\beta$ in $S_n\text{,}$ define $\alpha \sim \beta$ if there exists an $\sigma \in S_n$ such that $\sigma \alpha \sigma^{-1} = \beta\text{.}$ Show that $\sim$ is an equivalence relation on $S_n\text{.}$

###### 32.

Let $\sigma \in S_X\text{.}$ If $\sigma^n(x) = y$ for some $n \in \mathbb Z\text{,}$ we will say that $x \sim y\text{.}$

1. Show that $\sim$ is an equivalence relation on $X\text{.}$

2. Define the orbit of $x \in X$ under $\sigma \in S_X$ to be the set

\begin{equation*} {\mathcal O}_{x, \sigma} = \{ y : x \sim y \}\text{.} \end{equation*}

Compute the orbits of each element in $\{1, 2, 3, 4, 5\}$ under each of the following elements in $S_5\text{:}$

\begin{align*} \alpha & = (1254)\\ \beta & = (123)(45)\\ \gamma & = (13)(25)\text{.} \end{align*}
3. If ${\mathcal O}_{x, \sigma} \cap {\mathcal O}_{y, \sigma} \neq \emptyset\text{,}$ prove that ${\mathcal O}_{x, \sigma} = {\mathcal O}_{y, \sigma}\text{.}$ The orbits under a permutation $\sigma$ are the equivalence classes corresponding to the equivalence relation $\sim\text{.}$

4. A subgroup $H$ of $S_X$ is transitive if for every $x, y \in X\text{,}$ there exists a $\sigma \in H$ such that $\sigma(x) = y\text{.}$ Prove that $\langle \sigma \rangle$ is transitive if and only if ${\mathcal O}_{x, \sigma} = X$ for some $x \in X\text{.}$

###### 33.

Let $\alpha \in S_n$ for $n \geq 3\text{.}$ If $\alpha \beta = \beta \alpha$ for all $\beta \in S_n\text{,}$ prove that $\alpha$ must be the identity permutation; hence, the center of $S_n$ is the trivial subgroup.

###### 34.

If $\alpha$ is even, prove that $\alpha^{-1}$ is also even. Does a corresponding result hold if $\alpha$ is odd?

###### 35.

If $\sigma \in A_n$ and $\tau \in S_n\text{,}$ show that $\tau^{-1} \sigma \tau \in A_n\text{.}$

###### 36.

Show that $\alpha^{-1} \beta^{-1} \alpha \beta$ is even for $\alpha, \beta \in S_n\text{.}$

###### 37.

Let $r$ and $s$ be the elements in $D_n$ described in Theorem 5.21

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