 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 35623 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: 8. Contemporary Abstract Algebra was written by and is associated to the ISBN: 9781133599708.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

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.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Column space C (A) =
space of all combinations of the columns 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.

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.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

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

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).