 3.1CE: Exercise 1.This software determines the cyclic subgroups of U(n) (n...
 3.1E: For each group in the following list, find the order of the group a...
 3.2CE: Exercise 1.This software determines the cyclic subgroups of U(n) (n...
 3.2E: Let Q be the group of rational numbers under addition and let Q* be...
 3.3CE: Exercise 1.This software determines the cyclic subgroups of U(n) (n...
 3.3E: Let Q and Q* be as in Exercise 2. Find the order of each element in...
 3.4CE: Exercise 1.This software determines the cyclic subgroups of U(n) (n...
 3.4E: Prove that in any group, an element and its inverse have the same o...
 3.5CE: Exercise 1.This software determines the cyclic subgroups of U(n) (n...
 3.5E: Without actually computing the orders, explain why the two elements...
 3.6E: In the group Z12, find a, b, and a + b for each case.a. a = 6...
 3.7E: If a, b, and c are group elements and a = 6, b = 7, express (a4...
 3.8E: What can you say about a subgroup of D3 that contains R240 and a re...
 3.9E: What can you say about a subgroup of D4 that contains R270 and a re...
 3.10E: How many subgroups of order 4 does D4 have?
 3.11E: Determine all elements of finite order in R*, the group of nonzero ...
 3.12E: If a and b are group elements and ab ? ba, prove that aba ? e.
 3.13E: Suppose that H is a nonempty subset of a group G that is closed und...
 3.14E: Let G be the group of polynomials under addition with coefficients ...
 3.15E: If a is an element of a group G and a = 7, show that a is the cub...
 3.16E: Suppose that H is a nonempty subset of a group G with the property ...
 3.17E: Prove that if an Abelian group has more than three elements of orde...
 3.18E: Suppose that a is a group element and a6 = e. What are the possibil...
 3.19E: If a is a group element and a has infinite order, prove that am ? a...
 3.20E: Let x belong to a group. If x2 ? e and x6 = e, prove that x4 ? e an...
 3.21E: Show that if a is an element of a group G, then a ? G.
 3.22E: Show that . [Hence, U(14) is cyclic.] Is
 3.23E: Show that for any k in U(20). [Hence, U(20) is not cyclic.]
 3.24E: Suppose n is an even positive integer and H is a subgroup of Zn. Pr...
 3.25E: Prove that for every subgroup of Dn, either every member of the sub...
 3.26E: Prove that a group with two elements of order 2 that commute must h...
 3.27E: For every even integer n, show that Dn has a subgroup of order 4.
 3.28E: Suppose that H is a proper subgroup of Z under addition and H conta...
 3.29E: Suppose that H is a proper subgroup of Z under addition and that H ...
 3.30E: Prove that the dihedral group of order 6 does not have a subgroup o...
 3.31E: For each divisor k . 1 of n, let Uk(n) = {x ? U(n)  x mod k = 1}. ...
 3.32E: If H and K are subgroups of G, show that H ? K is a subgroup of G. ...
 3.33E: Let G be a group. Show that Z(G) = ?a?GC(a). [This means the inters...
 3.34E: Let G be a group, and let a ? G. Prove that C(a) = C(a–1).
 3.35E: For any group element a and any integer k, show that C(a) ? C(ak). ...
 3.36E: Complete the partial Cayley group table given below.
 3.37E: Suppose G is the group defined by the following Cayley table. a. Fi...
 3.38E: If a and b are distinct group elements, prove that either a2 ? b2 o...
 3.39E: Let S be a subset of a group and let H be the intersection of all s...
 3.40E: In the group Z, find
 3.41E: Prove Theorem 3.6.Theorem 3.6 C(a) Is a SubgroupFor each a in a gro...
 3.42E: If H is a subgroup of G, then by the centralizer C(H) of H we mean ...
 3.43E: Must the centralizer of an element of a group be Abelian?
 3.44E: Must the center of a group be Abelian?
 3.45E: Let G be an Abelian group with identity e and let n be some fixed i...
 3.46E: Suppose a belongs to a group and a = 5. Prove that C(a) = C(a3). ...
 3.47E: Let G be the set of all polynomials with coefficients from the set ...
 3.48E: In each case, find elements a and b from a group such that a =b...
 3.49E: Suppose a group contains elements a and b such that a = 4,b = 2...
 3.50E: Suppose a and b are group elements such that a = 2, b ? e, and ab...
 3.51E: Let a be a group element of order n, and suppose that d is a positi...
 3.52E: Consider the elements from SL(2, R). Find A, B, and AB. Does ...
 3.53E: Consider the element in SL(2, R). What is the order of A? If we vie...
 3.54E: For any positive integer n and any angle ?, show that in the group ...
 3.55E: Let G be the symmetry group of a circle. Show that G has elements o...
 3.56E: Let x belong to a group and x = 6. Find x2, x3, x4, and x5...
 3.57E: D 4 has seven cyclic subgroups. List them.
 3.58E: U(15) has six cyclic subgroups. List them.
 3.59E: Prove that a group of even order must have an element of order 2.
 3.60E: Suppose G is a group that has exactly eight elements of order 3. Ho...
 3.61E: Let H be a subgroup of a finite group G. Suppose that g belongs to ...
 3.62E: Compute the orders of the following groups.a. U(3), U(4), U(12)b. U...
 3.63E: Let R* be the group of nonzero real numbers under multiplication an...
 3.64E: Compute U(4), U(10), and U(40). Do these groups provide a cou...
 3.65E: Find a cyclic subgroup of order 4 in U(40).
 3.66E: Find a noncyclic subgroup of order 4 in U(40).
 3.67E: Let under addition. Let H = Prove that H is a subgroup of G. What i...
 3.68E: Let H = {A ? GL(2, R)  det A is an integer power of 2}. Show that ...
 3.69E: Let H be a subgroup of R under addition. Let K = {2a  a ? H}. Prov...
 3.70E: Let G be a group of functions from R to R*, where the operation of ...
 3.71E: Let G = GL(2, R) and nonzero integers under the operation of matrix...
 3.72E: Let H = {a + bi  a, b? R, ab ? 0}. Prove or disprove that H is a s...
 3.73E: Let H = {a + bi  a, b ? R, a2 + b2 = 1}. Prove or disprove that H ...
 3.74E: Let G be a finite Abelian group and let a and b belong to G. Prove ...
 3.75E: Let H be a subgroup of a group G. Prove that the set HZ(G) = {hz  ...
 3.76E: Let G be a group and H a subgroup. For any element g of G, define g...
 3.77E: Let a belong to a group and a = m. If n is relatively prime to m,...
 3.78E: Let F be a reflection in the dihedral group Dn and R a rotation in ...
 3.79E: Let G = GL(2, R).
 3.80E: Let G be a finite group with more than one element. Show that G has...
Solutions for Chapter 3: Contemporary Abstract Algebra 8th Edition
Full solutions for Contemporary Abstract Algebra  8th Edition
ISBN: 9781133599708
Solutions for Chapter 3
Get Full SolutionsSince 85 problems in chapter 3 have been answered, more than 43010 students have viewed full stepbystep solutions from this chapter. 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. Contemporary Abstract Algebra was written by and is associated to the ISBN: 9781133599708. Chapter 3 includes 85 full stepbystep solutions.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

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.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

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

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.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

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.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

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 ().

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

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

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.

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

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.