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Section5.4Exercises

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. \((1345)(234)\)

  2. \((12)(1253)\)

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

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

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

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

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

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

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

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

  11. \(|(1254)|\)

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

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

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

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

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

3

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

  1. \((14356)\)

  2. \((156)(234)\)

  3. \((1426)(142)\)

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

  5. \((142637)\)

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

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

  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 5.26, 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. \((1 2), (13), \ldots, (1n)\)

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

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

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 \}. \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 \(\sigma^n(x) = y\text{,}\) we will say that \(x \sim y\text{.}\)

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

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

  3. 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 \}. \end{equation*}

    Compute the orbits of each of the following elements in \(S_5\text{:}\)

    \begin{align*} \alpha & = (1254)\\ \beta & = (123)(45)\\ \gamma & = (13)(25). \end{align*}
  4. 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 \(\sim\text{.}\)

  5. 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

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

36

Let \(r\) and \(s\) be the elements in \(D_n\) described in Theorem 5.23

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

These exercises are designed to help you become familiar with permutation groups in Sage.

37Sage Exercise 1

Create the full symmetric group \(S_{10}\) with the command G = SymmetricGroup(10).

38Sage Exercise 2

Create elements of G with the following (varying) syntax. Pay attention to commas, quotes, brackets, parentheses. The first two use a string (characters) as input, mimicking the way we write permuations (but with commas). The second two use a list of tuples.

  • a = G("(5,7,2,9,3,1,8)")

  • b = G("(1,3)(4,5)")

  • c = G([(1,2),(3,4)])

  • d = G([(1,3),(2,5,8),(4,6,7,9,10)])

  1. Compute \(a^3\text{,}\) \(bc\text{,}\) \(ad^{-1}b\text{.}\)

  2. Compute the orders of each of these four individual elements (a through d) using a single permutation group element method.

  3. Use the permutation group element method .sign() to determine if \(a,b,c,d\) are even or odd permutations.

  4. Create two cyclic subgroups of \(G\) with the commands:

    • H = G.subgroup([a])

    • K = G.subgroup([d])

    List, and study, the elements of each subgroup. Without using Sage, list the order of each subgroup of \(K\text{.}\) Then use Sage to construct a subgroup of \(K\) with order 10.

  5. More complicated subgroups can be formed by using two or more generators. Construct a subgroup \(L\) of \(G\) with the command L = G.subgroup([b,c]). Compute the order of \(L\) and list all of the elements of \(L\text{.}\)

39Sage Exercise 3

Construct the group of symmetries of the tetrahedron (also the alternating group on 4 symbols, \(A_4\)) with the command A=AlternatingGroup(4). Using tools such as orders of elements, and generators of subgroups, see if you can find all of the subgroups of \(A_4\) (each one exactly once). Do this without using the .subgroups() method to justify the correctness of your answer (though it might be a convenient way to check your work).

Provide a nice summary as your answer—not just piles of output. So use Sage as a tool, as needed, but basically your answer will be a concise paragraph and/or table. This is the one part of this assignment without clear, precise directions, so spend some time on this portion to get it right. Hint: no subgroup of \(A_4\) requires more than two generators.

40Sage Exercise 4

The subsection The Motion Group of a Cube describes the \(24\) symmetries of a cube as a subgroup of the symmetric group \(S_8\) generated by three quarter-turns. Answer the following questions about this symmetry group.

  1. From the list of elements of the group, can you locate the ten rotations about axes? (Hint: the identity is easy, the other nine never send any symbol to itself.)

  2. Can you identify the six symmetries that are a transposition of diagonals? (Hint: [g for g in cube if g.order() == 2] is a good preliminary filter.)

  3. Verify that any two of the quarter-turns (above, front, right) are sufficient to generate the whole group. How do you know each pair generates the entire group?

  4. Can you express one of the diagonal transpositions as a product of quarter-turns? This can be a notoriously difficult problem, especially for software. It is known as the “word problem.”

  5. Number the six faces of the cube with the numbers \(1\) through \(6\) (any way you like). Now consider the same three symmetries we used before (quarter-turns about face-to-face axes), but now view them as permutations of the six faces. In this way, we construct each symmetry as an element of \(S_6\text{.}\) Verify that the subgroup generated by these symmetries is the whole symmetry group of the cube. Again, rather than using three generators, try using just two.

41Sage Exercise 5

Save your work, and then see if you can crash your Sage session by building the subgroup of \(S_{10}\) generated by the elements b and d of orders \(2\) and \(30\) from above. Do not submit the list of elements of N as part of your submitted worksheet.

What is the order of \(N\text{?}\)