 21.21.1: Justify each step in the proof that 1f] is a homomorphism in Exampl...
 21.21.2: Prove that 1f2 : A x B > B, as defined in Example 21.4, is a homo...
 21.21.3: Consider M (2, Z) as a group with respect to addition (Example 3.5 ...
 21.21.4: Find Ker Pr for p,. in Proble:n 21.3.
 21.21.5: Define e : Z6 + Z3 by e([a]6) = [ah for each [a]6 E Z6.(a) Prove t...
 21.21.6: (a) Show that ct : Z3 + Z6 given by ct([ah) = [a]6 is not well def...
 21.21.7: Prove that every homomorphic image of an Abelian group is Abelian.
 21.21.8: Prove that every homomorphic image of a cyclic group is cyclic.
 21.21.9: Provethatife : G + His a homomorphism and A is a subgroup ofG, the...
 21.21.10: Prove that if e : G + H is a homomorphism and B is a subgroup of H...
 21.21.11: Prove that if ct : G + H is a homomorphism and f3 : H + K is a ho...
 21.21.12: (a) With ct and f3 as in 21.11, prove that Ker ct S; Ker f3 0 ct.(b...
 21.21.13: Let nand k denote positive integers, and define e : Z + Z" byeta) ...
 21.21.14: Determine Ker e in each of the following cases, for e, n, and k as ...
 21.21.15: Let G denote the subgroup {1, 1, i, i} of complex numbers (operat...
 21.21.16: Define A,. from M(2, Z) to itself by A,. (x) = rx for each x E M(2,...
 21.21.17: Provethatife: G + His a homomorphism, a E G, and o(a) is finite, t...
 21.21.18: There is a unique homomorphism e : Z6 + S3 such that e([I]) = (1 2...
 21.21.19: True orfalse: If N <lG, then gnrl = n for all n E Nand g E G. Justi...
 21.21.20: Prove that N <l G iff gl ng E N for all n E Nand g E G. (In other ...
 21.21.21: By choosing a rectangular coordinate system, the points of a plane ...
 21.21.22: Determine all of the normal subgroups of the group of symmetries of...
 21.21.23: Prove that if & is a homomorphism from 0 onto H, and N <J 0, then &...
 21.21.24: If A and B are groups, then {e} X B <J A x B. Give two different pr...
 21.21.25: Prove that if C denotes any collection of normal subgroups of a gro...
 21.21.26: Prove that if N is a subgroup of 0, then N <J 0 iff Ng = gN for eac...
 21.21.27: Prove that if N is a subgroup of 0 and [0 : NJ = 2, then N <J O. (S...
 21.21.28: Determine all of the normal subgroups of S3. (See 17.12.)
 21.21.29: Prove that if Hand N are subgroups of a group 0 and N <J G, then H ...
 21.21.30: For C the complex numbers, let E, I, J, and K be the elements of M ...
 21.21.31: If G = (a) and e : 0 4 H is a homomorphism, then e is completely d...
 21.21.32: There is only one homomorphism from Z2 to Z3. Why?
 21.21.33: Verify that if 0 and H are any groups, and 8 : 0 4 H is defined by...
 21.21.34: Define&: Z x Z 4 Z by &0, b)) = a + b. Verify that 8 is a homomorp...
 21.21.35: Determine all of the homomorphisms of Z onto Z
Solutions for Chapter 21: HOMOMORPHISMS OF GROUPS. KERNELS
Full solutions for Modern Algebra: An Introduction  6th Edition
ISBN: 9780470384435
Solutions for Chapter 21: HOMOMORPHISMS OF GROUPS. KERNELS
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 21: HOMOMORPHISMS OF GROUPS. KERNELS includes 35 full stepbystep solutions. Modern Algebra: An Introduction was written by and is associated to the ISBN: 9780470384435. This textbook survival guide was created for the textbook: Modern Algebra: An Introduction, edition: 6. Since 35 problems in chapter 21: HOMOMORPHISMS OF GROUPS. KERNELS have been answered, more than 8139 students have viewed full stepbystep solutions from this chapter.

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!

Column space C (A) =
space of all combinations of the columns of A.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

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.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(DÂ» O.

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

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).

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Solvable system Ax = b.
The right side b is in the column space of A.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.