 9.1E: Let H = {(1), (12)}. Is H normal in S3?
 9.2E: Prove that An is normal in Sn.
 9.3E: In D4, let K = {R0, R90, R180, R270}. Write HR90 in the form xH, wh...
 9.4E: Write (12)(13)(14) in the form ? (12), where ? ? A4. Write (1234) (...
 9.5E: Show that if G is the internal direct product of H1, H2, . . . , Hn...
 9.6E: Let . Is H a normal subgroup of GL(2, R)?
 9.7E: Let G = GL(2, R) and let K be a subgroup of R*. Prove that H = {A ?...
 9.8E: Viewing is isomorphic to Z6. Generalize to arbitrary integers k and n.
 9.9E: Prove that if H has index 2 in G, then H is normal in G. (This exer...
 9.10E: Let H = {(1), (12)(34)} in A4.a. Show that H is not normal in A4.b....
 9.11E: Let . Show that G/H is not isomorphic to G/K. (This shows that H ? ...
 9.12E: Prove that a factor group of a cyclic group is cyclic.
 9.13E: Prove that a factor group of an Abelian group is Abelian.
 9.14E: What is the order of the element in the factor group
 9.15E: What is the order of the element 4U5(105) in the factor group U(105...
 9.16E: Recall that Z(D6) = {R0, R180}. What is the order of the element R6...
 9.17E: Let . List the elements of H and G/H.
 9.18E: What is the order of the factor group
 9.19E: What is the order of the factor group
 9.20E: Construct the Cayley table for U(20)/U5(20).
 9.21E: Prove that an Abelian group of order 33 is cyclic.
 9.22E: Determine the order of . Is the group cyclic?
 9.23E: Determine the order of . Is the group cyclic?
 9.24E: The group is isomorphic to one of Determine which one by elimination.
 9.25E: Let G = U(32) and H = {1, 31}. The group G/H is isomorphic to one o...
 9.26E: Let G be the group of quaternions given by the table in Exercise 4 ...
 9.27E: Let G = U(16), H = {1, 15}, and K={1, 9}. Are H and K isomorphic? A...
 9.28E: Let , H = {(0, 0), (2, 0), (0, 2), (2, 2)}, and . Is G/H isomorphic...
 9.29E: Prove that has no subgroup of order 18.
 9.30E: Express U(165) as an internal direct product of proper subgroups in...
 9.31E: Let R* denote the group of all nonzero real numbers under multiplic...
 9.32E: Prove that D4 cannot be expressed as an internal direct product of ...
 9.33E: Let H and K be subgroups of a group G. If G = HK and g = hk, where ...
 9.34E: In Z, let . Prove that Z = HK. Does Z = H × K?
 9.35E: Let G = {3a6b10c  a, b, c ?Z} under multiplication and H = {3a6b12...
 9.36E: Determine all subgroups of R* (nonzero reals under multiplication) ...
 9.37E: Let G be a finite group and let H be a normal subgroup of G. Prove ...
 9.38E: Let H be a normal subgroup of G and let a belong to G. If the eleme...
 9.39E: If H is a normal subgroup of a group G, prove that C(H), the centra...
 9.40E: Let ? be an isomorphism from a group G onto a group . Prove that if...
 9.41E: Show that Q, the group of rational numbers under addition, has no p...
 9.42E: An element is called a square if it can be expressed in the form b2...
 9.43E: Show, by example, that in a factor group G/H it can happen that aH ...
 9.44E: Observe from the table for A4 given in Table 5.1 on page 111 that t...
 9.45E: Let p be a prime. Show that if H is a subgroup of a group of order ...
 9.46E: Show that D13 is isomorphic to Inn(D13).
 9.47E: Suppose that N is a normal subgroup of a finite group G and H is a ...
 9.48E: If G is a group and G: Z(G) = 4, prove that .
 9.49E: Suppose that G is a nonAbelian group of order p3, where p is a pri...
 9.50E: If G = pq, where p and q are primes that are not necessarily dist...
 9.51E: Let N be a normal subgroup of G and let H be a subgroup of G. If N ...
 9.52E: Let G be an Abelian group and let H be the subgroup consisting of a...
 9.53E: Determine all subgroups of R* that have finite index.
 9.54E: Let G = {±1, ±i, ±j, ±k}, where i 2 = j2 = k2 = –1, –i = (–1)i, 12 ...
 9.55E: In D4, let K = {R0, D} and let L = {R0, D, D ', R180}. Show that bu...
 9.56E: Show that the intersection of two normal subgroups of G is a normal...
 9.57E: Give an example of subgroups H and K of a group G such that HK is n...
 9.58E: If N and M are normal subgroups of G, prove that NM is also a norma...
 9.59E: Let N be a normal subgroup of a group G. If N is cyclic, prove that...
 9.60E: Without looking at inner automorphisms of Dn, determine the number ...
 9.61E: Let H be a normal subgroup of a finite group G and let x?G. If gcd(...
 9.62E: Let G be a group and let G' be the subgroup of G generated by the s...
 9.63E: If N is a normal subgroup of G and G/N 5 m, show that xm ? N for ...
 9.64E: Suppose that a group G has a subgroup of order n. Prove that the in...
 9.65E: If G is nonAbelian, show that Aut(G) is not cyclic.
 9.66E: Let G = pnm, where p is prime and gcd( p, m) = 1. Suppose that H ...
 9.67E: Suppose that H is a normal subgroup of a finite group G. If G/H has...
 9.68E: Recall that a subgroup N of a group G is called characteristic if ?...
 9.69E: In D4, let Form an operation table for the cosets Is the result a g...
 9.70E: Prove that A4 is the only subgroup of S4 of order 12.
 9.71E: If G = 30 and Z(G) = 5, what is the structure of G/Z(G)?
 9.72E: If H is a normal subgroup of G and H = 2, prove that H is contain...
 9.73E: Prove that A5 cannot have a normal subgroup of order 2.
 9.74E: Let G be a finite group and let H be an oddorder subgroup of G of ...
 9.75E: Let G be a group and p a prime. Suppose that H = {gp Z g ? G} is a ...
 9.76E: Suppose that H is a normal subgroup of G. If H = 4 and gH has ord...
 9.77E: Let G be a group and H an oddorder subgroup of G of index 2. Show ...
 9.78E: A proper subgroup H of a group G is called maximal if there is no s...
 9.79E: Let G be a group of order 100 that has exactly one subgroup of orde...
Solutions for Chapter 9: Contemporary Abstract Algebra 8th Edition
Full solutions for Contemporary Abstract Algebra  8th Edition
ISBN: 9781133599708
Solutions for Chapter 9
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 79 problems in chapter 9 have been answered, more than 14369 students have viewed full stepbystep solutions from this chapter. Chapter 9 includes 79 full stepbystep solutions. This textbook survival guide was created for the textbook: Contemporary Abstract Algebra , edition: 8th. Contemporary Abstract Algebra was written by Sieva Kozinsky and is associated to the ISBN: 9781133599708.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

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

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

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.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

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

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

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.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.
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