 5.84E: What would be wrong with using the 2cycle notation (11) instead of...
 5.1CE: This software determines whether the two permutations (1x) and (123...
 5.1E: Let
 5.2E: Let
 5.3E: Write each of the following permutations as a product of disjoint c...
 5.4E: Find the order of each of the following permutations.a. (14)b. (147...
 5.5E: What is the order of each of the following permutations?a. (124)(35...
 5.6E: What is the order of each of the following permutations?
 5.7E: What is the order of the product of a pair of disjoint cycles of le...
 5.8E: Show that A8 contains an element of order 15.
 5.9E: What are the possible orders for the elements of S6 and A6? What ab...
 5.10E: What is the maximum order of any element in A10?
 5.11E: Determine whether the following permutations are even or odd.a. (13...
 5.12E: Show that a function from a finite set S to itself is onetoone if...
 5.13E: Suppose that a is a mapping from a set S to itself and ?(?(x)) = x ...
 5.14E: Find eight elements in S6 that commute with (12)(34)(56). Do they f...
 5.15E: Let n be a positive integer. If n is odd, is an ncycle an odd or a...
 5.16E: If ? is even, prove that ? –1 is even. If a is odd, prove that ? –1...
 5.17E: Prove Theorem 5.6.Theorem 5.6 Even Permutations Form a GroupThe set...
 5.18E: In Sn, let a be an rcycle, ? an scycle, and ? ? tcycle. Complete...
 5.19E: Let ? and ? belong to Sn. Prove that ?? is even if and only if ? an...
 5.20E: Associate an even permutation with the number +1 and an odd permuta...
 5.21E: Let ? be the permutation of the letters A through Z that takes each...
 5.22E: If ? and ? are distinct 2cycles, what are the possibilities for  ...
 5.23E: Show that if H is a subgroup of Sn, then either every member of H i...
 5.24E: Suppose that H is a subgroup of Sn of odd order. Prove that H is a ...
 5.25E: Give two reasons why the set of odd permutations in Sn is not a sub...
 5.26E: Let ? and ? belong to Sn. Prove that a–1b–1ab is an even permutation.
 5.27E: Use Table 5.1 to compute the following.a. The centralizer of ?3 5 (...
 5.28E: How many elements of order 5 are in S7?
 5.29E: How many elements of order 4 does S6 have? How many elements of ord...
 5.30E: Let ? ? S7 and suppose ?4 5 (2143567). Find ?. What are the possibi...
 5.31E: Let ? ? S7 and suppose ?4 5 (2143567). Find ?. What are the possibi...
 5.32E: Let ? = (123)(145). Write ?99 in disjoint cycle form.
 5.33E: Find three elements ? in S9 with the property that ? 3 = (157)(283)...
 5.34E: What cycle is (a1a2 … an)–1?
 5.35E: Let G be a group of permutations on a set X. Let a ? X and define s...
 5.36E: Let ? = (1,3,5,7,9,8,6)(2,4,10). What is the smallest positive inte...
 5.37E: Let ? = (1,3,5,7,9)(2,4,6)(8,10). If ?m is a 5cycle, what can you ...
 5.38E: Let H = {? ? S5  ?(1) = 1 and ?(3) = 3}. Prove that H is a subgrou...
 5.39E: How many elements of order 5 are there in A6?
 5.40E: In S4, find a cyclic subgroup of order 4 and a noncyclic subgroup o...
 5.41E: Suppose that ? is a 10cycle. For which integers i between 2 and 10...
 5.42E: In S3, find elements ? and ? such that a = 2, ? = 2, and ?? = 3.
 5.43E: Find group elements ? and ? in S5 such that ?  = 3,  ?  = 3, an...
 5.44E: Represent the symmetry group of an equilateral triangle as a group ...
 5.45E: Prove that Sn is nonAbelian for all n ? 3.
 5.46E: Prove that An is nonAbelian for all n ? 4.
 5.47E: For n ? 3, let H = { ? ? Sn  ? 1) = 1 or 2 and ? 2) = 1 or 2}. Pro...
 5.48E: Show that in S7, the equation x2 = (1234) has no solutions but the ...
 5.49E: If (ab) and (cd) are distinct 2cycles in Sn, prove that (ab) and (...
 5.50E: Let ? be a 2cycle and ? be a tcycle in Sn. Prove that ? ? ? is a ...
 5.51E: Use the previous exercise to prove that, if a and ? belong to Sn an...
 5.52E: Let ? and b belong to Sn. Prove that ?? ? –1 and a are both even or...
 5.53E: What is the smallest positive integer n such that Sn has an element...
 5.54E: Let n be an even positive integer. Prove that An has an element of ...
 5.55E: Let n be an odd positive integer. Prove that An has an element of o...
 5.56E: Let n be an even positive integer. Prove that An has an element of ...
 5.57E: Viewing the members of D4 as a group of permutations of a square la...
 5.58E: Viewing the members of D5 as a group of permutations of a regular p...
 5.59E: Let n be an odd integer greater than 1. Viewing Dn as a group of pe...
 5.60E: Let n be an integer greater than 1. Viewing Dn as a group of permut...
 5.61E: Show that A5 has 24 elements of order 5, 20 elements of order 3, an...
 5.62E: Find a cyclic subgroup of A8 that has order 4.
 5.63E: Find a noncyclic subgroup of A8 that has order 4.
 5.64E: Compute the order of each member of A4. What arithmetic relationshi...
 5.65E: Show that every element in An for n ? 3 can be expressed as a 3cyc...
 5.66E: Show that for n ? 3, Z(Sn) 5 {e}.
 5.67E: Verify the statement made in the discussion of the Verhoeff check d...
 5.68E: Use the Verhoeff checkdigit scheme based on D5 to append a check d...
 5.69E: Prove that every element of Sn (n > 1) can be written as a product ...
 5.70E: (Indiana College Mathematics Competition) A cardshuffling machine ...
 5.71E: Show that a permutation with odd order must be an even permutation.
 5.72E: Let G be a group. Prove or disprove that H = {g2  g ? G} is a subg...
 5.73E: Let H = {a2  a ? S4} and K = {a2  a ? S5}. Prove H = A4 and K = A5.
 5.74E: Let H = {a2  a ? S6}. Prove H ? A6.
 5.75E: Determine integers n for which H = {? ? An  ?2 = ?} is a subgroup ...
 5.76E: Given that ? and ? are in S4 with ?? = 114322, ?? = (1243), and ?(1...
 5.77E: Why does the fact that the orders of the elements of A4 are 1, 2, a...
 5.78E: Find five subgroups of S5 of order 24.
 5.79E: Find six subgroups of order 60 in S6.
 5.80E: For n > 1, let H be the set of all permutations in Sn that can be e...
 5.81E: Shown below are four tire rotation patterns recommended by the Dunl...
 5.82E: Label the four locations of tires on an automobile with the labels ...
 5.83E: What would be wrong with using the 2cycle notation (11) instead of...
Solutions for Chapter 5: Permutation Groups
Full solutions for Contemporary Abstract Algebra  8th Edition
ISBN: 9781133599708
Solutions for Chapter 5: Permutation Groups
Get Full SolutionsSummary of Chapter 5: Permutation Groups
In this chapter, we study certain groups of functions, called permutation groups, from a set A to itself.
This expansive textbook survival guide covers the following chapters and their solutions. Contemporary Abstract Algebra was written by and is associated to the ISBN: 9781133599708. Chapter 5: Permutation Groups includes 85 full stepbystep solutions. Since 85 problems in chapter 5: Permutation Groups have been answered, more than 238747 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Contemporary Abstract Algebra , edition: 8.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.