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# 5.4: Exercises

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## 1

Write the following permutations in cycle notation.

1. $\begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 2 & 4 & 1 & 5 & 3 \end{pmatrix} \nonumber$
2. $\begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 4 & 2 & 5 & 1 & 3 \end{pmatrix} \nonumber$
3. $\begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 3 & 5 & 1 & 4 & 2 \end{pmatrix} \nonumber$
4. $\begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 1 & 4 & 3 & 2 & 5 \end{pmatrix} \nonumber$

## 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

$Z(G) = \{ g \in G : gx = xg \text{ for all } x \in G \}\text{.} \nonumber$

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

$\sigma \tau \sigma^{-1 } = ( \sigma(a_1), \sigma(a_2), \ldots, \sigma(a_k)) \nonumber$

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

${\mathcal O}_{x, \sigma} = \{ y : x \sim y \}\text{.} \nonumber$

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

This page titled 5.4: Exercises is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by Thomas W. Judson (Abstract Algebra: Theory and Applications) via source content that was edited to the style and standards of the LibreTexts platform.

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