 8.1CE: This software lists the elements of Us(st), where s and t are relat...
 8.1E: Prove that the external direct product of any finite number of grou...
 8.1SE: A subgroup N of a group G is called a characteristic subgroup if ?(...
 8.2CE: Exercise 2. This software computes the elements of the subgroup U(n...
 8.2E: Show that Z2 ? Z2 ? Z2 has seven subgroups of order 2.
 8.2SE: Prove that the center of a group is characteristic.
 8.3CE: Exercise 2. This software computes the elements of the subgroup U(n...
 8.3E: Let G be a group with identity eG and let H be a group with identit...
 8.3SE: The commutator subgroup G' of a group G is the subgroup generated b...
 8.4CE: This software allows you to input positive integers n_1, n_2>, n_3,...
 8.4E: Show that G ? H is Abelian if and only if G and H are Abelian. Stat...
 8.4SE: Prove that the property of being a characteristic subgroup is trans...
 8.5CE: Exercise 2. This software computes the elements of the subgroup U(n...
 8.5E: Prove or disprove that Z ? Z is a cyclic group.
 8.5SE: Let and let H be the subgroup of SL(3, Z3) consisting of (See Exerc...
 8.6CE: This software determines the order of , where p is a prime less tha...
 8.6E: Prove, by comparing orders of elements, that Z8 ?Z2 is not isomorph...
 8.6SE: Let H and K be subgroups of a group G and let HK = {hk  h?H, k?K} ...
 8.7CE: This software determines the order of , where p is a prime less tha...
 8.7E: Prove that G1 ? G2 is isomorphic to G2 ? G1. State the general case.
 8.7SE: Let G be a finite Abelian group in which every nonidentity element ...
 8.8CE: This software determines the order of , where p is a prime less tha...
 8.8E: Is Z3 ?Z9 isomorphic to Z27? Why?
 8.8SE: Prove that S4 is not isomorphic to .
 8.9CE: This software determines the order of , where p is a prime less tha...
 8.9E: Is Z3 ? Z5 isomorphic to Z15? Why?
 8.9SE: Let G be a group. For any element g of G, define gZ(G) = {gh  h ? ...
 8.10CE: This software determines the order of , where p is a prime less tha...
 8.10E: How many elements of order 9 does Z3 ? Z9 have? (Do not do this exe...
 8.10SE: The exponent of a group is the smallest positive integer n such tha...
 8.11E: How many elements of order 4 does Z4 ? Z4 have? (Do not do this by ...
 8.11SE: Determine all Ugroups of exponent 2.
 8.12E: Give examples of four groups of order 12, no two of which are isomo...
 8.12SE: Suppose that H and K are subgroups of a group and that H and K ...
 8.13E: For each integer n > 1, give examples of two nonisomorphic groups o...
 8.13SE: Let R+ denote the multiplicative group of positive real numbers and...
 8.14E: The dihedral group Dn of order 2n (n ? 3) has a subgroup of n rotat...
 8.14SE: Prove that Q* under multiplication is not isomorphic to R* under mu...
 8.15E: Prove that the group of complex numbers under addition is isomorphi...
 8.15SE: Prove that Q under addition is not isomorphic to R under addition.
 8.16E: Suppose that G1 ? G2 and H1 ? H2. Prove that G1 ? H1 ? G2 ? H2. Sta...
 8.16SE: Prove that R under addition is not isomorphic to R* under multiplic...
 8.17E: If G ? H is cyclic, prove that G and H are cyclic. State the genera...
 8.17SE: Show that Q+ (the set of positive rational numbers) under multiplic...
 8.18E: In Z40 ? Z30, find two subgroups of order 12.
 8.18SE: Suppose that G = {e, x, x2, y, yx, yx2} is a nonAbelian group with...
 8.19E: If r is a divisor of m and s is a divisor of n, find a subgroup of ...
 8.19SE: Let p be an odd prime. Show that 1 is the only solution of xp–2 = 1...
 8.20E: Find a subgroup of Z12 ? Z18 that is isomorphic to Z9 ? Z4.
 8.20SE: Let G be an Abelian group under addition. Let n be a fixed positive...
 8.21E: Let G and H be finite groups and (g, h) ?G ? H. State a necessary a...
 8.21SE: Find a subgroup of .
 8.22E: Determine the number of elements of order 15 and the number of cycl...
 8.22SE: Suppose that . Prove that Z(G) =
 8.23E: What is the order of any nonidentity element of Z3 ? Z3 ? Z3? Gener...
 8.23SE: Exhibit four nonisomorphic groups of order 18.
 8.24E: Let m >2 be an even integer and let n > 2 be an odd integer. Find a...
 8.24SE: What is the order of the largest cyclic subgroup in Aut(Z720)? (Hin...
 8.25E: Let M be the group of all real 2 × 2 matrices under addition. Let N...
 8.25SE: Let G be the group of all permutations of the positive integers. Le...
 8.26E: The group S3 ? Z2 is isomorphic to one of the following groups: Z12...
 8.26SE: Let G be a group and let g ? G. Show that is a subgroup of G.
 8.27E: Let G be a group, and let H = {(g, g)  g?G}. Show that H is a subg...
 8.27SE: Show that (This exercise is referred to in Chapter 24.)
 8.28E: Find a subgroup of Z4 ? Z2 that is not of the form H ? K, where H i...
 8.28SE: Show that . (This exercise is referred to in Chapter 24.)
 8.29E: Find all subgroups of order 3 in Z9 ? Z3.
 8.29SE: Show that . (This exercise is referred to in Chapter 24.)
 8.30E: Find all subgroups of order 4 in Z4 ? Z4.
 8.30SE: Exhibit four nonisomorphic groups of order 66. (This exercise is re...
 8.31E: What is the largest order of any element in Z30 ? Z20?
 8.31SE: Prove that Inn(G) = 1 if and only if G is Abelian.
 8.32E: What is the order of the largest cyclic subgroup of Z6 ? Z10 ? Z15?...
 8.32SE: Prove that x100 = 1 for all x in U(1000).
 8.33E: Find three cyclic subgroups of maximum possible order in Z6 ? Z10 ?...
 8.33SE: Find a subgroup of order 6 in U(450).
 8.34E: How many elements of order 2 are in Z2000000 ? Z4000000? Generalize.
 8.34SE: List four elements of that form a noncyclic subgroup.
 8.35E: Find a subgroup of Z800 ? Z200 that is isomorphic to Z2 ? Z4.
 8.35SE: In S10, let ? = (13)(17)(265)(289). Find an element in S10 that com...
 8.36E: Find a subgroup of Z12 ? Z4 ? Z15 that has order 9.
 8.36SE: Prove or disprove that
 8.37E: Prove that R* ? R* is not isomorphic to C*. (Compare this with Exer...
 8.37SE: Prove or disprove that
 8.38E: Let See Exercise 48 in Chapter 2 for the definition of multiplicati...
 8.38SE: Describe a threedimensional solid whose symmetry group is isomorph...
 8.39E: Let G = {3m6n  m, n ? Z} under multiplication. Prove that G is iso...
 8.39SE: Let . Find the order of (2, 3, (123)(15)). Find the inverse of (2, ...
 8.40E: Let (a1, a2, . . . , an) ? G1 ? G2 ? … ? Gn. Give a necessary and s...
 8.40SE: Let and let H = {g ? G  g = ? or g = 1}. Prove or disprove tha...
 8.41E: Prove that D3 ? D4 ? D12? Z2.
 8.41SE: Find a subgroup H of is isomorphic to
 8.42E: Determine the number of cyclic subgroups of order 15 in Z90 ? Z36. ...
 8.42SE: Find three subgroups H1, H2, and H3 of such that ( )/Hi is isomorph...
 8.43E: List the elements in the groups U5(35) and U7(35).
 8.43SE: Find an element of order 10 in A9.
 8.44E: Prove or disprove that U(40) ? Z6 is isomorphic to U(72) ? Z4.
 8.44SE: In the left regular representation for D4, write TR90 and TH in mat...
 8.45E: Prove or disprove that C* has a subgroup isomorphic to Z2 ? Z2.
 8.45SE: How many elements of order 6 are in S7?
 8.46E: Let G be a group isomorphic to Zn1 ? Zn2 ? . . . ? Znk. Let x be th...
 8.46SE: Prove that is not isomorphic to a subgroup of S6.
 8.47E: If a group has exactly 24 elements of order 6, how many cyclic subg...
 8.47SE: Find a permutation ? such that ?2 = (13579)(268).
 8.48E: For any Abelian group G and any positive integer n, let Gn = {gn  ...
 8.48SE: In under component wise addition, let H = {(x, 3x)  x ? R}. (Note ...
 8.49E: Express Aut(U(25)) in the form Zm ? Zn.
 8.49SE: In , suppose that H is the subgroup of all points lying on a line t...
 8.50E: Determine Aut(Z2 ? Z2).
 8.50SE: Let G be a group of permutations on the set {1, 2, . . . , n}. Reca...
 8.51E: Suppose that n1, n2, . . . , nk are positive even integers. How man...
 8.51SE: Let H be a subgroup of G and let a, b ? G. Show that aH = bH if and...
 8.52E: Is Z10 ? Z12 ? Z6 ? Z60 ? Z6 ? Z2?
 8.52SE: Suppose that G is a finite Abelian group that does not contain a su...
 8.53E: Is Z10 ? Z12 ? Z6 ? Z15 ? Z4 ? Z12?
 8.53SE: Let p be a prime. Determine the number of elements of order p in
 8.54E: Find an isomorphism from Z12 to Z4 ? Z3.
 8.54SE: Show that .
 8.55E: How many isomorphisms are there from Z12 to Z4 ? Z3?
 8.55SE: Let p be a prime. Determine the number of subgroups of that are iso...
 8.56E: Suppose that ? is an isomorphism from Z3 ? Z= to Z15 and ? (2, 3) 5...
 8.56SE: Find a group of order that contains a subgroup isomorphic to A8.
 8.57E: If f is an isomorphism from Z4 ? Z3 to Z12, what is ? (2, 0)? What ...
 8.57SE: Let p and q be distinct odd primes. Let n = lcm(p – 1, q – 1). Prov...
 8.58E: Prove that Z5 ? Z5 has exactly six subgroups of order 5.
 8.58SE: Give a simple characterization of all positive integers n for which...
 8.59E: Let (a, b) belong to Zm ? Zn. Prove that (a, b) divides lcm(m, n).
 8.59SE: Prove that the permutations (12) and (123 . . . n) generate Sn. (Th...
 8.60E: Let G = {ax2 1 bx 1 c  a, b, c ? Z3}. Add elements of G as you wou...
 8.60SE: Suppose that n is even and s is an (n – 1)cycle in Sn. Show that s...
 8.61E: Determine all cyclic groups that have exactly two generators.
 8.61SE: Suppose that n is odd and ? is an ncycle in Sn. Prove that ? does ...
 8.62E: Explain a way that a string of length n of the four nitrogen bases ...
 8.62SE: Let H = {? ? Sn  a maps the set {1, 2} to itself}. Prove that C ((...
 8.63E: Let p be a prime. Prove that Zp ? Zp has exactly p + 1 subgroups of...
 8.63SE: Let m be a positive integer. For any ncycle ?, show that ?m is the...
 8.64E: Give an example of an infinite nonAbelian group that has exactly s...
 8.65E: Give an example to show that there exists a group with elements a a...
 8.66E: Express U(165) as an external direct product of cyclic groups of th...
 8.67E: Express U(165) as an external direct product of Ugroups in four di...
 8.68E: Without doing any calculations in Aut(Z20), determine how many elem...
 8.69E: Without doing any calculations in Aut(Z720), determine how many ele...
 8.70E: Without doing any calculations in U(27), decide how many subgroups ...
 8.71E: What is the largest order of any element in U(900)?
 8.72E: Let p and q be odd primes and let m and n be positive integers. Exp...
 8.73E: Use the results presented in this chapter to prove that U(55) is is...
 8.74E: Use the results presented in this chapter to prove that U(144) is i...
 8.75E: For every n > 2, prove that U(n)2 = {x2  x ? U(n)} is a proper sub...
 8.76E: Show that U(55)3 = {x3  x ? U(55)} is U(55).
 8.77E: Find an integer n such that U(n) contains a subgroup isomorphic to ...
 8.78E: Find a subgroup of order 6 in U(700).
 8.79E: Show that there is a Ugroup containing a subgroup isomorphic to Z3...
 8.80E: Find an integer n such that U(n) is isomorphic to Z2 ? Z4 ? Z9.
 8.81E: What is the smallest positive integer k such that xk = e for all x ...
 8.82E: If k divides m and m divides n, how are Um(n) and Uk(n) related?
 8.83E: Let p1, p2,…, pk be distinct odd primes and n1, n2,…, nk be positiv...
 8.84E: Show that no Ugroup has order 14.
 8.85E: Show that there is a Ugroup containing a subgroup isomorphic to Z14.
 8.86E: Show that no Ugroup is isomorphic to Z4 ? Z4.
 8.87E: Show that there is a Ugroup containing a subgroup isomorphic to Z4...
 8.88E: Using the RSA scheme with p = 37, q = 73, and e = 5, what number wo...
 8.89E: Assuming that a message has been sent via the RSA scheme with p = 3...
Solutions for Chapter 8: Contemporary Abstract Algebra 8th Edition
Full solutions for Contemporary Abstract Algebra  8th Edition
ISBN: 9781133599708
Solutions for Chapter 8
Get Full SolutionsChapter 8 includes 162 full stepbystep solutions. Since 162 problems in chapter 8 have been answered, more than 51016 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their 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.

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

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

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!

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

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

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.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

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).

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

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.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

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.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

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

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

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·