## 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. $$\displaystyle (1 \, 3 \, 4 \, 5)(2 \, 3 \, 4)$$

2. $$\displaystyle (1 \, 2)(1 \, 2 \, 5 \, 3)$$

3. $$\displaystyle (1 \, 4 \, 3)(2 \, 3)(2 \, 4)$$

4. $$\displaystyle (1 \, 4 \, 2 \, 3)(3 \, 4)(5 \, 6)(1 \, 3 \, 2 \, 4)$$

5. $$\displaystyle (1 \, 2 \, 5 \, 4)(1 \, 3)(2 \, 5)$$

6. $$\displaystyle (1 \, 2 \, 5 \, 4) (1 \, 3)(2 \, 5)^2$$

7. $$\displaystyle (1 \, 2 \, 5 \, 4)^{-1} (1 \, 2 \, 3)(4 \, 5) (1 \, 2 \, 5 \, 4)$$

8. $$\displaystyle (1 \, 2 \, 5 \, 4)^2 (1 \, 2 \, 3)(4 \, 5)$$

9. $$\displaystyle (1 \, 2 \, 3)(4 \, 5) (1 \, 2 \, 5 \, 4)^{-2}$$

10. $$\displaystyle (1 \, 2 \, 5 \, 4)^{100}$$

11. $$\displaystyle |(1 \, 2 \, 5 \, 4)|$$

12. $$\displaystyle |(1 \, 2 \, 5 \, 4)^2|$$

13. $$\displaystyle (1 \, 2)^{-1}$$

14. $$\displaystyle (1 \, 2 \, 5 \, 3 \, 7)^{-1}$$

15. $$\displaystyle [(1 \, 2)(3 \, 4)(1 \, 2)(4 \, 7)]^{-1}$$

16. $$\displaystyle [(1 \, 2 \, 3 \, 5)(4 \, 6 \, 7)]^{-1}$$

###### 3.

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

1. $$\displaystyle (1 \, 4 \, 3 \, 5 \, 6)$$

2. $$\displaystyle (1 \, 5 \, 6)(2 \, 3 \, 4)$$

3. $$\displaystyle (1 \, 4 \, 2 \, 6)(1 \, 4 \, 2)$$

4. $$\displaystyle (1 \, 7 \, 2 \, 5 \, 4)(1 \, 4 \, 2 \, 3)(1 \, 5 \, 4 \, 6 \, 3 \, 2)$$

5. $$\displaystyle (1 \, 4 \, 2 \, 6 \, 3 \, 7)$$

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

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

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

###### 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 & = (1 \, 2 \, 5 \, 4)\\ \beta & = (1 \, 2 \, 3)(4 \, 5)\\ \gamma & = (1 \, 3)(2 \, 5)\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 $$n / \gcd(k,n)\text{.}$$