- 22.22.1: Determine the order of each of the following quotient groups. (a) Z...
- 22.22.2: Determine the order of each of the following quotient groups. (a) Z...
- 22.22.3: Construct the Cayley table for Z12/([4J). (Suggestion: Use [[klJ to...
- 22.22.4: Construct the Cayley table for the group in Example 22.3.
- 22.22.5: Prove that every quotient group of an Abelian group is Abelian.
- 22.22.6: Prove that every quotient group of a cyclic group is cyclic.
- 22.22.7: If m and n are positive integers and min, then (n) is a normal subg...
- 22.22.8: Give a reason for each step in the proof of Theorem 22.1 for why th...
- 22.22.9: Assume N <l G.(a) Prove that if [G : N] is a prime, then G / N is c...
- 22.22.10: Determine the order of (Z12 x Z4)/, )). Explain the differenc...
- 22.22.11: Prove that if N <l G and a E G, then o(N a) I o(a). [Here o(N a) de...
- 22.22.12: Prove that every element of Q/Z has finite order. Also show that Q/...
- 22.22.13: The elements of finite order in an Abelian group form a subgroup ( ...
- 22.22.14: Prove that G / N is Abelian iff aba-I b-I E N for all a, bEG.
- 22.22.15: Prove that if N is a subgroup of G, and the operation (Na)(Nb) = N(...
Solutions for Chapter 22: QUOTIENT GROUPS
Full solutions for Modern Algebra: An Introduction | 6th Edition
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!
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.
peA) = det(A - AI) has peA) = zero matrix.
Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).
Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.
Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.
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.
Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.
A symmetric matrix with eigenvalues of both signs (+ and - ).
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.
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 •
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).
Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.
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.
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.
Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.