 26.1E: Let S be a set of distinct symbols. Show that the relation defined ...
 26.2E: Let n be an even integer. Prove that Dn/Z(Dn) is isomorphic to Dn/2.
 26.3E: Verify that the set K in Example 2 is closed under multiplication o...
 26.4E: Show that is isomorphic to Z2.
 26.5E: Prove Theorem 26.3 and its corollary.
 26.6E: Let G be the group {±1, ±i, ±j, ±k} with multiplication defined as ...
 26.7E: In any group, show that . (This exercise is referred to in the proo...
 26.8E: Let ? = (12)(34) and ? = (24). Show that the group generated by ? a...
 26.9E: Prove that is isomorphic to Dn.(This exercise is referred to in the...
 26.10E: What is the minimum number of generators needed for Z2 ? Z2 ? Z2? F...
 26.11E: Suppose that x2 = y2 = e and yz = zxy. Show that xy = yx.
 26.12E: Let a. Express a3b2abab3 in the form bia j, where 0 ? i ? 1 and 0 ?...
 26.13E: Let a. Express b2abab3 in the form biaj.b. Express b3abab3a in the ...
 26.14E: Let G be the group defined by the following table. Show that G is i...
 26.15E: Let . Show that G ? 16. Assuming that G = 16, find the center o...
 26.16E: Confirm the classification given in Table 26.1 of all groups of ord...
 26.17E: Let G be defined by some set of generators and relations. Show that...
 26.18E: Let Show that the permutations (23) and (13) satisfy the defining r...
 26.19E: In (which is isomorphic to D4) is not a normal subgroup.
 26.20E: Let . Show that Z(G) = {e, xn}. Assuming that G = 4n, show that G...
 26.21E: Let . How many elements does G have? To what familiar group is G is...
 26.22E: Let Show that G ? 16. Assuming that G = 16, find the center of ...
 26.23E: Determine the orders of the elements of D?.
 26.24E: Let Prove that G is isomorphic to D4.
 26.25E: Let . Determine G.
 26.26E: Let . To what familiar group is G isomorphic?
 26.27E: Let . To what familiar group is G isomorphic?
 26.28E: Give an example of a nonAbelian group that has exactly three eleme...
 26.29E: Referring to Example 7 in this chapter, show as many letters as you...
 26.30E: Suppose that a group of order 8 has exactly five elements of order ...
Solutions for Chapter 26: Contemporary Abstract Algebra 8th Edition
Full solutions for Contemporary Abstract Algebra  8th Edition
ISBN: 9781133599708
Solutions for Chapter 26
Get Full SolutionsThis 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. Chapter 26 includes 30 full stepbystep solutions. Since 30 problems in chapter 26 have been answered, more than 43010 students have viewed full stepbystep solutions from this chapter. Contemporary Abstract Algebra was written by and is associated to the ISBN: 9781133599708.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

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.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

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.

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

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.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

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.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = 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.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

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

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 ().

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

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.