- 58.58.1: What are the orders of the Sylow p-subgroups of a group of order 180?
- 58.58.2: What are the orders of the Sylow p-subgroups of a group of order 700?
- 58.58.3: Verify Lemma 58.1 by direct computation for 3
- 58.58.4: Verify Lemma 58.1 by direct computation for4
- 58.58.5: Find all of the Sylow p-subgroups of 53 for p = 2 and p = 3. Verify...
- 58.58.6: Use Sylow's Theorem (Extended Version) to verify that if IGI = 24, ...
- 58.58.7: Explain why every group of order 12 must have a subgroup of every o...
- 58.58.8: Assume that n and r are integers such that 0 < r < n.(a) Explain wh...
- 58.58.9: Assume that p is a prime.(a) Prove that if 0 < j ~ pk - I, then p' ...
- 58.58.10: (a) Verify that if H is a subgroup of G, and a E G, then aHa- 1 is ...
- 58.58.11: (a) Use Sylow's Theorem (Extended Version) to prove that every grou...
- 58.58.12: Prove that there is no simple group of order 20. (See Example 58.1.)
- 58.58.13: Prove that a group of order 175 must have normal subgroups of order...
- 58.58.14: Let G be a group of order 56.(a) Say as much as you can, based on S...
- 58.58.15: Prove that if p is a prime and G is a finite group such that pk IIG...
Solutions for Chapter 58: SYLOW'S THEOREM
Full solutions for Modern Algebra: An Introduction | 6th Edition
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).
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
A = CTC = (L.J]))(L.J]))T for positive definite A.
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.
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.
Gram-Schmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.
Inverse matrix A-I.
Square matrix with A-I A = I and AA-l = 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 B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
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 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.
Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.
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).
R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().
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.
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.
Tridiagonal matrix T: tij = 0 if Ii - j I > 1.
T- 1 has rank 1 above and below diagonal.