 7.1CE: This software determines when Zn is the only group of order n in th...
 7.1E: Let H = {(1), (12)(34), (13)(24), (14)(23)}. Find the left cosets o...
 7.2E: Let H be as in Exercise 1. How many left cosets of H in S4 are ther...
 7.3E: Let H = {0, ±3, ±6, ±9, . . .}. Find all the left cosets of H in Z.
 7.4E: Rewrite the condition a–1b ? H given in property 5 of the lemma on ...
 7.5E: Let H be as in Exercise 3. Use Exercise 4 to decide whether or not ...
 7.6E: Let n be a positive integer. Let H = {0, ±n, ±2n, ±3n, . . .}. Find...
 7.7E: Find all of the left cosets of {1, 11} in U(30).
 7.8E: Suppose that a has order 15. Find all of the left cosets of
 7.9E: Let a = 30. How many left cosets of are there? List them.
 7.10E: Give an example of a group G and subgroups H and K such that HK = {...
 7.11E: If H and K are subgroups of G and g belongs to G, show that g(H ? K...
 7.12E: Let a and b be nonidentity elements of different orders in a group ...
 7.13E: Let H be a subgroup of R*, the group of nonzero real numbers under ...
 7.14E: Let C* be the group of nonzero complex numbers under multiplication...
 7.15E: Let G be a group of order 60. What are the possible orders for the ...
 7.16E: Suppose that K is a proper subgroup of H and H is a proper subgroup...
 7.17E: Let G be a group with G = pq, where p and q are prime. Prove that...
 7.18E: Recall that, for any integer n greater than 1, ?(n) denotes the num...
 7.19E: Compute 515 mod 7 and 713 mod 11.
 7.20E: Use Corollary 2 of Lagrange’s Theorem (Theorem 7.1) to prove that t...
 7.21E: Suppose G is a finite group of order n and m is relatively prime to...
 7.22E: Suppose H and K are subgroups of a group G. If H = 12 and K = 3...
 7.23E: Suppose that H is a subgroup of S4 and that H contains (12) and (23...
 7.24E: Suppose that H and K are subgroups of G and there are elements a an...
 7.25E: Suppose that G is an Abelian group with an odd number of elements. ...
 7.26E: Suppose that G is a group with more than one element and G has no p...
 7.27E: Let G = 15. If G has only one subgroup of order 3 and only one of...
 7.28E: Let G be a group of order 25. Prove that G is cyclic or g5 = e for ...
 7.29E: Let G = 33. What are the possible orders for the elements of G? S...
 7.30E: Let G = 8. Show that G must have an element of order 2.
 7.31E: Can a group of order 55 have exactly 20 elements of order 11? Give ...
 7.32E: Determine all finite subgroups of C*, the group of nonzero complex ...
 7.33E: Let H and K be subgroups of a finite group G with H?K?G. Prove that...
 7.34E: Suppose that a group contains elements of orders 1 through 10. What...
 7.35E: Give an example of the dihedral group of smallest order that contai...
 7.36E: Show that in any group of order 100, either every element has order...
 7.37E: Suppose that a finite Abelian group G has at least three elements o...
 7.38E: Prove that if G is a finite group, the index of Z(G) cannot be prime.
 7.39E: Find an example of a subgroup H of a group G and elements a and b i...
 7.40E: Prove that a group of order 63 must have an element of order 3.
 7.41E: Let G be a group of order 100 that has a subgroup H of order 25. Pr...
 7.42E: Let G be a group of order n and k be any integer relatively prime t...
 7.43E: Let G be a group of permutations of a set S. Prove that the orbits ...
 7.44E: Prove that every subgroup of Dn of odd order is cyclic.
 7.45E: Let G = {(1), (12)(34), (1234)(56), (13)(24), (1432)(56), (56)(13),...
 7.46E: Prove that a group of order 12 must have an element of order 2.
 7.47E: Show that in a group G of odd order, the equation x2 = a has a uniq...
 7.48E: Let G be a group of order pqr, where p, q, and r are distinct prime...
 7.49E: Prove that a group that has more than one subgroup of order 5 must ...
 7.50E: Prove that A5 has a subgroup of order 12.
 7.51E: Prove that A5 has no subgroup of order 30.
 7.52E: Prove that A5 has no subgroup of order 15 to 20.
 7.53E: Suppose that a is an element from a permutation group G and one of ...
 7.54E: Let G be a group and suppose that H is a subgroup of G with the pro...
 7.55E: Prove that A5 is the only subgroup of S5 of order 60.
 7.56E: Why does the fact that A4 has no subgroup of order 6 imply that Z(...
 7.57E: Let G = GL(2, R) and H = SL(2, R). Let A?G and suppose that det A =...
 7.58E: Let G be the group of rotations of a plane about a point P in the p...
 7.59E: Let G be the rotation group of a cube. Label the faces of the cube ...
 7.60E: The group D4 acts as a group of permutations of the square regions ...
 7.61E: Let G = GL(2, R), the group of 2 × 2 matrices over R with nonzero d...
 7.62E: Calculate the orders of the following (refer to Figure 27.5 for ill...
 7.63E: Prove that the eightelement set in the proof of Theorem 7.5 is a g...
 7.64E: A soccer ball has 20 faces that are regular hexagons and 12 faces t...
 7.65E: If G is a finite group with fewer than 100 elements and G has subgr...
Solutions for Chapter 7: Contemporary Abstract Algebra 8th Edition
Full solutions for Contemporary Abstract Algebra  8th Edition
ISBN: 9781133599708
Solutions for Chapter 7
Get Full SolutionsChapter 7 includes 66 full stepbystep solutions. Contemporary Abstract Algebra was written by and is associated to the ISBN: 9781133599708. This textbook survival guide was created for the textbook: Contemporary Abstract Algebra , edition: 8. This expansive textbook survival guide covers the following chapters and their solutions. Since 66 problems in chapter 7 have been answered, more than 42560 students have viewed full stepbystep solutions from this chapter.

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

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

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.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

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.

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).

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

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).

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

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

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.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.