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Contents Index Dark Mode Prev Up Next \(\newcommand{\identity}{\mathrm{id}}
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Exercises 5.4 Exercises
1.
Write the following permutations in cycle notation.
\begin{equation*}
\begin{pmatrix}
1 & 2 & 3 & 4 & 5 \\
2 & 4 & 1 & 5 & 3
\end{pmatrix}
\end{equation*}
\begin{equation*}
\begin{pmatrix}
1 & 2 & 3 & 4 & 5 \\
4 & 2 & 5 & 1 & 3
\end{pmatrix}
\end{equation*}
\begin{equation*}
\begin{pmatrix}
1 & 2 & 3 & 4 & 5 \\
3 & 5 & 1 & 4 & 2
\end{pmatrix}
\end{equation*}
\begin{equation*}
\begin{pmatrix}
1 & 2 & 3 & 4 & 5 \\
1 & 4 & 3 & 2 & 5
\end{pmatrix}
\end{equation*}
2.
Compute each of the following.
\(\displaystyle (1 \, 3 \, 4 \, 5)(2 \, 3 \, 4)\)
\(\displaystyle (1 \, 2)(1 \, 2 \, 5 \, 3)\)
\(\displaystyle (1 \, 4 \, 3)(2 \, 3)(2 \, 4)\)
\(\displaystyle (1 \, 4 \, 2 \, 3)(3 \, 4)(5 \, 6)(1 \, 3 \, 2 \, 4)\)
\(\displaystyle (1 \, 2 \, 5 \, 4)(1 \, 3)(2 \, 5)\)
\(\displaystyle (1 \, 2 \, 5 \, 4) (1 \, 3)(2 \, 5)^2\)
\(\displaystyle (1 \, 2 \, 5 \, 4)^{-1} (1 \, 2 \, 3)(4 \, 5) (1 \, 2 \, 5 \, 4)\)
\(\displaystyle (1 \, 2 \, 5 \, 4)^2 (1 \, 2 \, 3)(4 \, 5)\)
\(\displaystyle (1 \, 2 \, 3)(4 \, 5) (1 \, 2 \, 5 \, 4)^{-2}\)
\(\displaystyle (1 \, 2 \, 5 \, 4)^{100}\)
\(\displaystyle |(1 \, 2 \, 5 \, 4)|\)
\(\displaystyle |(1 \, 2 \, 5 \, 4)^2|\)
\(\displaystyle (1 \, 2)^{-1}\)
\(\displaystyle (1 \, 2 \, 5 \, 3 \, 7)^{-1}\)
\(\displaystyle [(1 \, 2)(3 \, 4)(1 \, 2)(4 \, 7)]^{-1}\)
\(\displaystyle [(1 \, 2 \, 3 \, 5)(4 \, 6 \, 7)]^{-1}\)
3.
Express the following permutations as products of transpositions and identify them as even or odd.
\(\displaystyle (1 \, 4 \, 3 \, 5 \, 6)\)
\(\displaystyle (1 \, 5 \, 6)(2 \, 3 \, 4)\)
\(\displaystyle (1 \, 4 \, 2 \, 6)(1 \, 4 \, 2)\)
\(\displaystyle (1 \, 7 \, 2 \, 5 \, 4)(1 \, 4 \, 2 \, 3)(1 \, 5 \, 4 \, 6 \, 3 \, 2)\)
\(\displaystyle (1 \, 4 \, 2 \, 6 \, 3 \, 7)\)
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:
\(\displaystyle \{ \sigma \in S_4 : \sigma(1) = 3 \}\)
\(\displaystyle \{ \sigma \in S_4 : \sigma(2) = 2 \}\)
\(\{ \sigma \in S_4 : \sigma(1) = 3\) 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.
\(\displaystyle (1 \, 2), (1 \, 3), \ldots, (1 \, n)\)
\(\displaystyle (1 \, 2), (2 \, 3), \ldots, (n- 1,n)\)
\(\displaystyle (1 \, 2), (1 \, 2 \ldots n )\)
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 \(k\text{.}\)
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 \(k\text{.}\)
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 \(x \sim y\text{.}\)
Show that
\(\sim\) is an equivalence relation on
\(X\text{.}\)
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*}
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
\(\sim\text{.}\)
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
Show that
\(srs = r^{-1}\text{.}\)
Show that
\(r^k s = s r^{-k}\) in
\(D_n\text{.}\)
Prove that the order of
\(r^k \in D_n\) is
\(n / \gcd(k,n)\text{.}\)
