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Permutation Groups: exercises
Permutation Groups: exercises — from Judson, Abstract Algebra: Theory and Applications.
Practice (37)
Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.
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Write the following permutations in cycle notation.
\[\begin{aligned}\end{aligned}\]
\[\begin{aligned}\end{aligned}\]
\[\begin{aligned}\end{aligned}\]
\[\begin{aligned}\end{aligned}\]
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
Hint:
(a) \((1 \, 2 \, 4 \, 5 \, 3)\); (c) \((1 \, 3)(2 \, 5)\).
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Compute each of the following.
\((1 \, 3 \, 4 \, 5)(2 \, 3 \, 4)\)
\((1 \, 2)(1 \, 2 \, 5 \, 3)\)
\((1 \, 4 \, 3)(2 \, 3)(2 \, 4)\)
\((1 \, 4 \, 2 \, 3)(3 \, 4)(5 \, 6)(1 \, 3 \, 2 \, 4)\)
\((1 \, 2 \, 5 \, 4)(1 \, 3)(2 \, 5)\)
\((1 \, 2 \, 5 \, 4) (1 \, 3)(2 \, 5)^2\)
\((1 \, 2 \, 5 \, 4)^{-1} (1 \, 2 \, 3)(4 \, 5) (1 \, 2 \, 5 \, 4)\)
\((1 \, 2 \, 5 \, 4)^2 (1 \, 2 \, 3)(4 \, 5)\)
\((1 \, 2 \, 3)(4 \, 5) (1 \, 2 \, 5 \, 4)^{-2}\)
\((1 \, 2 \, 5 \, 4)^{100}\)
\(|(1 \, 2 \, 5 \, 4)|\)
\(|(1 \, 2 \, 5 \, 4)^2|\)
\((1 \, 2)^{-1}\)
\((1 \, 2 \, 5 \, 3 \, 7)^{-1}\)
\([(1 \, 2)(3 \, 4)(1 \, 2)(4 \, 7)]^{-1}\)
\([(1 \, 2 \, 3 \, 5)(4 \, 6 \, 7)]^{-1}\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
Hint:
(a) \((1 \, 3 \, 5)(2 \, 4)\); (c) \((1 \, 4)(2 \, 3)\); (e) \((1 \, 3 \, 2 \, 4)\); (g) \((1 \, 3 \, 4)(2 \, 5)\); (n) \((1 \, 7 \, 3 \, 5 \, 2)\).
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Express the following permutations as products of transpositions and identify them as even or odd.
\((1 \, 4 \, 3 \, 5 \, 6)\)
\((1 \, 5 \, 6)(2 \, 3 \, 4)\)
\((1 \, 4 \, 2 \, 6)(1 \, 4 \, 2)\)
\((1 \, 7 \, 2 \, 5 \, 4)(1 \, 4 \, 2 \, 3)(1 \, 5 \, 4 \, 6 \, 3 \, 2)\)
\((1 \, 4 \, 2 \, 6 \, 3 \, 7)\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
Hint:
(a) \((1 \, 6)(1 \, 5)(1 \, 3)(1 \, 4)\); (c) \((1 \, 6)(1 \, 4)(1 \, 2)\).
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Find \((a_1, a_2, \ldots, a_n)^{-1}\).
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
Hint:
\((a_1, a_2, \ldots, a_n)^{-1} = (a_1, a_{n}, a_{n-1}, \ldots, a_2)\)
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List all of the subgroups of \(S_4\). Find each of the following sets:
\(\{ \sigma \in S_4 : \sigma(1) = 3 \}\)
\(\{ \sigma \in S_4 : \sigma(2) = 2 \}\)
\(\{ \sigma \in S_4 : \sigma(1) = 3\) and \(\sigma(2) = 2 \}\).
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
Hint:
(a) \(\{ (1 \, 3), (1 \, 3)(2 \, 4), (1 \, 3 \, 2), (1 \, 3 \, 4), (1 \, 3 \, 2 \, 4), (1 \, 3 \, 4 \, 2) \}\) is not a subgroup.
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Find all of the subgroups in \(A_4\). What is the order of each subgroup?
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Find all possible orders of elements in \(S_7\) and \(A_7\).
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Show that \(A_{10}\) contains an element of order \(15\).
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
Hint:
\((1 \, 2 \, 3 \, 4 \, 5)(6 \, 7 \, 8)\).
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Does \(A_8\) contain an element of order \(26\)?
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Find an element of largest order in \(S_n\) for \(n = 3, \ldots, 10\).
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What are the possible cycle structures of elements of \(A_5\)? What about \(A_6\)?
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
Hint:
Permutations of the form \[\begin{aligned}\end{aligned}\] are possible for \(A_5\).
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Let \(\sigma \in S_n\) have order \(n\). Show that for all integers \(i\) and \(j\), \(\sigma^i = \sigma^j\) if and only if \(i \equiv j \pmod{n}\).
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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 \(\sigma_1, \ldots, \sigma_m\).
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Using cycle notation, list the elements in \(D_5\). What are \(r\) and \(s\)? Write every element as a product of \(r\) and \(s\).
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If the diagonals of a cube are labeled as , to which motion of the cube does the permutation \((12)(34)\) correspond? What about the other permutations of the diagonals?
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Find the group of rigid motions of a tetrahedron. Show that this is the same group as \(A_4\).
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Prove that \(S_n\) is nonabelian for \(n \geq 3\).
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
Hint:
Calculate \((1 \, 2 \, 3)(1 \, 2)\) and \((1 \, 2)(1 \, 2 \, 3)\).
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Show that \(A_n\) is nonabelian for \(n \geq 4\).
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Prove that \(D_n\) is nonabelian for \(n \geq 3\).
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Let \(\sigma \in S_n\) be a cycle. Prove that \(\sigma\) can be written as the product of at most \(n-1\) transpositions.
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Let \(\sigma \in S_n\). If \(\sigma\) is not a cycle, prove that \(\sigma\) can be written as the product of at most \(n - 2\) transpositions.
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If \(\sigma\) can be expressed as an odd number of transpositions, show that any other product of transpositions equaling \(\sigma\) must also be odd.
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If \(\sigma\) is a cycle of odd length, prove that \(\sigma^2\) is also a cycle.
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Show that a \(3\)-cycle is an even permutation.
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Prove that in \(A_n\) with \(n \geq 3\), any permutation is a product of cycles of length \(3\).
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
Hint:
Consider the cases \((a,b)(b,c)\) and \((a,b)(c,d)\).
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Prove that any element in \(S_n\) can be written as a finite product of the following permutations.
\((1 \, 2), (1 \, 3), \ldots, (1 \, n)\)
\((1 \, 2), (2 \, 3), \ldots, (n- 1,n)\)
\((1 \, 2), (1 \, 2 \ldots n )\)
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Let \(G\) be a group and define a map \(\lambda_g : G \rightarrow G\) by \(\lambda_g(a) = g a\). Prove that \(\lambda_g\) is a permutation of \(G\).
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Prove that there exist \(n!\) permutations of a set containing \(n\) elements.
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Recall that the center of a group \(G\) is \[\begin{aligned}\end{aligned}\]. Find the center of \(D_8\). What about the center of \(D_{10}\)? What is the center of \(D_n\)?
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
Hint:
Show that the center of \(D_n\) consists of the identity if \(n\) is odd and consists of the identity and a \(180^\circ\) rotation if \(n\) is even.
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Let \(\tau = (a_1, a_2, \ldots, a_k)\) be a cycle of length \(k\).
Prove that if \(\sigma\) is any permutation, then \[\begin{aligned}\end{aligned}\] is a cycle of length \(k\).
Let \(\mu\) be a cycle of length \(k\). Prove that there is a permutation \(\sigma\) such that \(\sigma \tau \sigma^{-1 } = \mu\).
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
Hint:
For (a), show that \(\sigma \tau \sigma^{-1 }(\sigma(a_i)) = \sigma(a_{i + 1})\).
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For \(\alpha\) and \(\beta\) in \(S_n\), define \(\alpha \sim \beta\) if there exists an \(\sigma \in S_n\) such that \(\sigma \alpha \sigma^{-1} = \beta\). Show that \(\sim\) is an equivalence relation on \(S_n\).
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Let \(\sigma \in S_X\). If \(\sigma^n(x) = y\) for some \(n \in \mathbb Z\), we will say that \(x \sim y\).
Show that \(\sim\) is an equivalence relation on \(X\).
Define the orbit of \(x \in X\) under \(\sigma \in S_X\) to be the set \[\begin{aligned}\end{aligned}\]. Compute the orbits of each element in \(\{1, 2, 3, 4, 5\}\) under each of the following elements in \(S_5\): \[\begin{aligned}\alpha & = (1 \, 2 \, 5 \, 4) \\ \beta & = (1 \, 2 \, 3)(4 \, 5) \\ \gamma & = (1 \, 3)(2 \, 5)\end{aligned}\].
If \({\mathcal O}_{x, \sigma} \cap {\mathcal O}_{y, \sigma} \neq \emptyset\), prove that \({\mathcal O}_{x, \sigma} = {\mathcal O}_{y, \sigma}\). The orbits under a permutation \(\sigma\) are the equivalence classes corresponding to the equivalence relation \(\sim\).
A subgroup \(H\) of \(S_X\) is transitive if for every \(x, y \in X\), there exists a \(\sigma \in H\) such that \(\sigma(x) = y\). Prove that \(\langle \sigma \rangle\) is transitive if and only if \({\mathcal O}_{x, \sigma} = X\) for some \(x \in X\).
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Let \(\alpha \in S_n\) for \(n \geq 3\). If \(\alpha \beta = \beta \alpha\) for all \(\beta \in S_n\), prove that \(\alpha\) must be the identity permutation; hence, the center of \(S_n\) is the trivial subgroup.
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If \(\alpha\) is even, prove that \(\alpha^{-1}\) is also even. Does a corresponding result hold if \(\alpha\) is odd?
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If \(\sigma \in A_n\) and \(\tau \in S_n\), show that \(\tau^{-1} \sigma \tau \in A_n\).
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Show that \(\alpha^{-1} \beta^{-1} \alpha \beta\) is even for \(\alpha, \beta \in S_n\).
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Let \(r\) and \(s\) be the elements in \(D_n\) described in
Show that \(srs = r^{-1}\).
Show that \(r^k s = s r^{-k}\) in \(D_n\).
Prove that the order of \(r^k \in D_n\) is \(n / \gcd(k,n)\).
Symbols used here
n divides a − b; a and b have the same remainder.
x belongs to A; every element of A is in B.
Typical distance from the mean; its square.
i² = −1.
Both signs at once: x = 3 ± 2 means 5 and 1.
Inequalities that allow equality; < and > exclude it.
Naturals, integers, rationals, reals, complex numbers.
Marks the point where the statement has been established.
b is a multiple of a; the largest number dividing both.
A set with an operation; the do-nothing element; the element that undoes g.
Same structure; the group of cosets of a normal subgroup N.
The remainders 0…n−1 with clock arithmetic.
The set of morphisms; do g then f.
Questions people ask
What is a group, in plain words?
A set with one operation that is associative, has an identity, and lets every element be undone. Symmetries of any object form a group — that is where the idea came from.
What is the difference between a ring and a field?
A ring has addition and multiplication that behave like the integers (you cannot always divide); a field is a ring where every non-zero element has a reciprocal, like the rationals or the reals.
ನಿಮ್ಮದೇ ಆದದ್ದನ್ನು ಪ್ರಯತ್ನಿಸಿ
Parts of this page are adapted from Judson, Abstract Algebra: Theory and Applications (GFDL 1.3). Condensed and re-explained here; errors are ours.
ಇನ್ನಷ್ಟು Abstract Algebra
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