 4.1CE: Exercise 1. This software determines if U(n) is cyclic. Run the pro...
 4.1E: Find all generators of Z6, Z8, and Z20.
 4.1SE: Let G be a group and let H be a subgroup of G. For any fixed x in G...
 4.2CE: Exercise 1. This software determines if U(n) is cyclic. Run the pro...
 4.2E: Suppose that are cyclic groups of orders 6, 8, and 20, respectively...
 4.2SE: Let G be a group and let H be a subgroup of G. Define N(H) = {x?G ...
 4.3CE: Exercise 1. This software determines if U(n) is cyclic. Run the pro...
 4.3E: List the elements of the subgroups in Z30. Let a be a group element...
 4.3SE: Let G be a group. For each a?G, define cl(a) 5 {xax–1  x?G}. Prove...
 4.4CE: Exercise 1. This software determines if U(n) is cyclic. Run the pro...
 4.4E: List the elements of the subgroups in Z18. Let a be a group element...
 4.4SE: The group defined by the following table is called the group of qua...
 4.5CE: Exercise 1. This software determines if U(n) is cyclic. Run the pro...
 4.5E: List the elements of the subgroups in U(20).
 4.5SE: (Conjugation preserves order.) Prove that, in any group, xax1 5 ...
 4.6CE: Exercise 1. This software determines if U(n) is cyclic. Run the pro...
 4.6E: What do Exercises 3, 4, and 5 have in common? Try to make a general...
 4.6SE: Prove that, in any group, ab = ba.
 4.7E: Find an example of a noncyclic group, all of whose proper subgroups...
 4.7SE: If a and b are group elements, prove that ab = a1b1.
 4.8E: Let a be an element of a group and let a = 15. Compute the orders...
 4.8SE: Prove that a group of order 4 cannot have a subgroup of order 3.
 4.9E: How many subgroups does Z20 have? List a generator for each of thes...
 4.9SE: If a, b, and c are elements of a group, give an example to show tha...
 4.10E: In Z24, list all generators for the subgroup of order 8. Let G = an...
 4.10SE: Let a and b belong to a group G. Prove that there is an element x i...
 4.11E: Let G be a group and let a ?G. Prove that
 4.11SE: Prove that if a is the only element of order 2 in a group, then a l...
 4.12E: In Z, find all generators of the subgroup If a has infinite order, ...
 4.12SE: Let G be the plane symmetry group of the infinite strip of equally ...
 4.13E: In Z24, find a generator for . Suppose that a = 24. Find a genera...
 4.13SE: What are the orders of the elements of D15? How many elements have ...
 4.14E: Suppose that a cyclic group G has exactly three subgroups: G itself...
 4.14SE: Prove that a group of order 4 is Abelian.
 4.15E: Let G be an Abelian group and let H = {g ? G  g divides 12}. Pro...
 4.15SE: Prove that a group of order 5 must be cyclic.
 4.16E: Find a collection of distinct subgroups of Z240 with the property t...
 4.16SE: Prove that an Abelian group of order 6 must be cyclic.
 4.17E: Complete the following statement: a = a2 if and only if a . ....
 4.17SE: Let G be an Abelian group and let n be a fixed positive integer. Le...
 4.18E: If a cyclic group has an element of infinite order, how many elemen...
 4.18SE: Let , where a and b are rational numbers not both 0. Prove that G i...
 4.19E: List the cyclic subgroups of U(30).
 4.19SE: (1969 Putnam Competition) Prove that no group is the union of two p...
 4.20E: Suppose that G is an Abelian group of order 35 and every element of...
 4.20SE: Prove that the subset of elements of finite order in an Abelian gro...
 4.21E: Let G be a group and let a be an element of G.a. If a12 = e, what c...
 4.21SE: Let p be a prime and let G be an Abelian group. Show that the set o...
 4.22E: Prove that a group of order 3 must be cyclic.
 4.22SE: Suppose that a and b are group elements. If b = 2 and bab = a4, d...
 4.23E: Let Z denote the group of integers under addition. Is every subgrou...
 4.23SE: Suppose that a finite group is generated by two elements a and b (t...
 4.24E: For any element a in any group G, prove that is a subgroup of C(a) ...
 4.24SE: If a is an element from a group and a = n, prove that C(a) = C(ak...
 4.25E: If d is a positive integer, d ? 2, and d divides n, show that the n...
 4.25SE: Let x and y belong to a group G. If xy?Z(G), prove that xy= yx.
 4.26E: Find all generators of Z. Let a be a group element that has infinit...
 4.26SE: Suppose that H and K are nontrivial subgroups of Q under addition. ...
 4.27E: Prove that C*, the group of nonzero complex numbers under multiplic...
 4.27SE: Let H be a subgroup of G and let g be an element of G. Prove that N...
 4.28E: Let a be a group element that has infinite order. Prove that if and...
 4.28SE: Let H be a subgroup of a group G and let g = n. If gm belongs to ...
 4.29E: List all the elements of order 8 in Z8000000. How do you know your ...
 4.29SE: Find a group that contains elements a and b such that a = 2, b ...
 4.30E: Suppose a and b belong to a group, a has odd order, and aba–1 = b–1...
 4.30SE: Suppose that G is a group with exactly eight elements of order 10. ...
 4.31E: Let G be a finite group. Show that there exists a fixed positive in...
 4.31SE: (1989 Putnam Competition) Let S be a nonempty set with an associati...
 4.32E: Determine the subgroup lattice for Z12.
 4.32SE: Let H1, H2, H3, . . . be a sequence of subgroups of a group with th...
 4.33E: Determine the subgroup lattice for Zp2q, where p and q are distinct...
 4.33SE: Let n be an integer greater than 1. Find a noncyclic subgroup of U ...
 4.34E: Determine the subgroup lattice for Z8.
 4.34SE: Let G be an Abelian group and H = {x ?G  xn = e for some odd integ...
 4.35E: Determine the subgroup lattice for Zpn, where p is a prime and n is...
 4.35SE: Let H = {A?GL(2, R)  det A is rational}. Prove or disprove that H ...
 4.36E: Prove that a finite group is the union of proper subgroups if and o...
 4.36SE: Suppose that G is a group that has exactly one nontrivial proper su...
 4.37E: Show that the group of positive rational numbers under multiplicati...
 4.37SE: Suppose that G is a group and G has exactly two nontrivial proper s...
 4.38E: Consider the set {4, 8, 12, 16}. Show that this set is a group unde...
 4.38SE: If a2= b2, prove or disprove that a = b.
 4.39E: Give an example of a group that has exactly 6 subgroups (including ...
 4.39SE: (1995 Putnam Competition) Let S be a set of real numbers that is cl...
 4.40E: Let m and n be elements of the group Z. Find a generator for the group
 4.40SE: If p is an odd prime, prove that there is no group that has exactly...
 4.41E: Suppose that a and b are group elements that commute and have order...
 4.41SE: Give an example of a group G with infinitely many distinct subgroup...
 4.42E: Suppose that a and b belong to a group G, a and b commute, and a ...
 4.42SE: Suppose a and b are group elements and b ? e. If a–1ba = b2 and a...
 4.43E: Suppose that a and b belong to a group G, a and b commute, and a ...
 4.43SE: Let a and b belong to a group G. Show that there is an element g in...
 4.44E: Let F and F' be distinct reflections in D21. What are the possibili...
 4.44SE: Suppose G is a group and x3y3 = y3x3 for every x and y in G. Let H ...
 4.45E: Suppose that H is a subgroup of a group G and H = 10. If a belong...
 4.45SE: Let G be a finite group and let S be a subset of G that contains mo...
 4.46E: Which of the following numbers could be the exact number of element...
 4.46SE: Let G be a group and let f be a function from G to some set. Show t...
 4.47E: If G is an infinite group, what can you say about the number of ele...
 4.47SE: Let G be a cyclic group of order n and let H be the subgroup of ord...
 4.48E: Suppose that K is a proper subgroup of D35 and K contains at least ...
 4.48SE: Let a be an element of maximum order from a finite Abelian group G....
 4.49E: For each positive integer n, prove that C*, the group of nonzero co...
 4.49SE: Define an operation * on the set of integers by a * b = a+ b + 1. S...
 4.50E: Prove or disprove that H = {n ? Z  n is divisible by both 8 and 10...
 4.50SE: Let n be an integer greater than 1. Find a noncyclic subgroup of U(...
 4.51E: Suppose that G is a finite group with the property that every nonid...
 4.52E: Prove that an infinite group must have an infinite number of subgro...
 4.53E: Let p be a prime. If a group has more than p – 1 elements of order ...
 4.54E: Suppose that G is a cyclic group and that 6 divides G. How many e...
 4.55E: List all the elements of Z40 that have order 10. Let x = 40. List...
 4.56E: Reformulate the corollary of Theorem 4.4 to include the case when t...
 4.57E: Determine the orders of the elements of D33 and how many there are ...
 4.58E: If G is a cyclic group and 15 divides the order of G, determine the...
 4.59E: If G is an Abelian group and contains cyclic subgroups of orders 4 ...
 4.60E: If G is an Abelian group and contains cyclic subgroups of orders 4 ...
 4.61E: Prove that no group can have exactly two elements of order 2.
 4.62E: Given the fact that U(49) is cyclic and has 42 elements, deduce the...
 4.63E: Let a and b be elements of a group. If a = 10 and b = 21, show ...
 4.64E: Let a and b belong to a group. If a and b are relatively prime,...
 4.65E: Let a and b belong to a group. If a = 24 and b = 10, what are t...
 4.66E: Prove that U(2n) (n ? 3) is not cyclic.
 4.67E: Suppose that G is a group of order 16 and that, by direct computati...
 4.68E: Prove that Zn has an even number of generators if n > 2. What does ...
 4.69E: If a5 = 12, what are the possibilities for a? If a4 = 12, wha...
 4.70E: Suppose that x = n. Find a necessary and sufficient condition on ...
 4.71E: Suppose a is a group element such that a28 = 10 and a22 = 20. D...
 4.72E: Let a be a group element such that a = 48. For each part, find a ...
 4.73E: Let p be a prime. Show that in a cyclic group of order pn –1, every...
 4.74E: Prove that is a cyclic subgroup of GL(2, R).
 4.75E: Let a and b belong to a group. If a = 12, b = 22, and {e}, prov...
 4.76E: (2008 GRE Practice Exam) If x is an element of a cyclic group of or...
 4.77E: Determine the number of cyclic subgroups of order 4 in Dn.
 4.78E: If n is odd, prove that Dn has no subgroup of order 4.
 4.79E: If n ? 4 and is even, show that Dn has exactly n/2 noncyclic subgro...
 4.80E: If n ? 4 and n is divisible by 2 but not by 4, prove that Dn has ex...
 4.81E: How many subgroups of order n does Dn have?
 4.82E: Let G be the set of all polynomials of the form ax2 + bx + c with c...
 4.83E: Let a and b belong to some group. Suppose that a = m, b = n, an...
 4.84E: For every integer n greater than 2, prove that the group U(n2 = 1) ...
 4.85E: Prove that for any prime p and positive integer n,?(pn) = pn – pn–1.
 4.86E: Give an example of an infinite group that has exactly two elements ...
Solutions for Chapter 4: Contemporary Abstract Algebra 8th Edition
Full solutions for Contemporary Abstract Algebra  8th Edition
ISBN: 9781133599708
Solutions for Chapter 4
Get Full SolutionsSince 142 problems in chapter 4 have been answered, more than 17602 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 4 includes 142 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.

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

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

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

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

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

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.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

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

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

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

Solvable system Ax = b.
The right side b is in the column space of A.

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

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.
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