 17.17.1: Find [S3 : ((1 2))].
 17.17.2: Find [S4 : ((1 2 3))].
 17.17.3: Find [ZIO : ([2])].
 17.17.4: Find [Z40 : ([12], [20])].
 17.17.5: Let G denote the group in Example 8.1 (the group of symmetries of a...
 17.17.6: With G as in 17.5, find [G : (J.L270)].
 17.17.7: A group G has subgroups of orders 4 and 10, and I G I < 50. What ca...
 17.17.8: A finite group G has elements of orders p and q, where p and q are ...
 17.17.9: Assume that G is a group with a subgroup H such that I H I = 6, [G ...
 17.17.10: Assume thatGis a group with a subgroupH such that I G I < 45, I HI>...
 17.17.11: Find all of the subgroups of Z6. Also construct the subgroup lattice.
 17.17.12: Find all of the subgroups of S3. Also construct the subgroup lattice.
 17.17.13: Find all of the subgroups of Z2 x Z2.
 17.17.14: Find all of the subgroups of Z3 x Z3.
 17.17.15: Determine the number of subgroups of Zp x Zp if P is a prime. (Comp...
 17.17.16: Find all of the subgroups of Z2 x ~. (There are eight).
 17.17.17: Find all of the subgroups of Z36. Also construct the subgroup latti...
 17.17.18: Assume that G is a cyclic group of order n, that G = (a), that kin,...
 17.17.19: Assume that H is a subgroup of a finite group G, and that G contain...
 17.17.20: Assume that A is a subgroup of a finite group G and that B is a sub...
 17.17.21: Assume that A is a finite group and let D denote the diagonal subgr...
 17.17.22: The exponent of a group G is the smallest positive integer n such t...
 17.17.23: Determine the exponent of each of the following groups. (See 17.22....
 17.17.24: Prove that if G is a group of order p2 (p a prime) and G is not cyc...
 17.17.25: Prove that ifH is a subgroup ofG, [G : H] = 2, a, bEG, a 'I H, and ...
 17.17.26: Verify that S4 has at least one subgroup of order k for each diviso...
 17.17.27: Prove that if A and 8 are finite subgroups of a group G, and IAI an...
 17.17.28: The subgroup A4 = (I 2 3), (1 2)(3 4) of S4 has order 12. Determine...
 17.17.29: Prove that a finite cyclic group of order n has exactly one subgrou...
 17.17.30: If H is a subgroup of G and [G : H] = 2, then the right cosets of H...
 17.17.31: Write a proof of Lagrange's Theorem using left cosets rather than r...
 17.17.32: Prove that if H is a subgroup of a finite group G, then the number ...
 17.17.33: Prove that if G and H are cyclic groups of orders m and n, with (m,...
 17.17.34: Prove that if Hand K are subgroups of a finite group G, and K f; H,...
Solutions for Chapter 17: LAGRANGE'S THEOREM. CYCLIC GROUPS
Full solutions for Modern Algebra: An Introduction  6th Edition
ISBN: 9780470384435
Solutions for Chapter 17: LAGRANGE'S THEOREM. CYCLIC GROUPS
Get Full SolutionsModern Algebra: An Introduction was written by and is associated to the ISBN: 9780470384435. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Modern Algebra: An Introduction, edition: 6. Since 34 problems in chapter 17: LAGRANGE'S THEOREM. CYCLIC GROUPS have been answered, more than 8298 students have viewed full stepbystep solutions from this chapter. Chapter 17: LAGRANGE'S THEOREM. CYCLIC GROUPS includes 34 full stepbystep solutions.

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.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

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.

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.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

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

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

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.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

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.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

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.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.