Contemporary Abstract Algebra
9th Edition
ISBN: 9781305657960
Author: Joseph Gallian
Publisher: Cengage Learning
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Textbook Question
Chapter 5, Problem 8E
Determine whether the following permutations are even or odd.
a. (135)
b. (1356)
c. (13567)
d. (12)(134)(152)
e. (1243)(3521)
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Question 4
(a) The following matrices represent linear maps on R² with respect to an
orthonormal basis:
=
[1/√5 2/√5
[2/√5 -1/√5]
"
[1/√5 2/√5]
A =
B =
[2/√5 1/√5] 1
C =
D =
=
=
[ 1/3/5 2/35]
1/√5 2/√5
-2/√5 1/√5'
For each of the matrices A, B, C, D, state whether it represents a self-adjoint
linear map, an orthogonal linear map, both, or neither.
(b) For the quadratic form
q(x, y, z) = y² + 2xy +2yz
over R, write down a linear change of variables to u, v, w such that q in these
terms is in canonical form for Sylvester's Law of Inertia.
[6]
[4]
part b please
Question 5
(a) Let a, b, c, d, e, ƒ Є K where K is a field. Suppose that the determinant of the
matrix
a cl
|df
equals 3 and the determinant of
determinant of the matrix
a+3b cl
d+3e f
ГЪ
e
[ c ] equals 2. Compute the
[5]
(b) Calculate the adjugate Adj (A) of the 2 × 2 matrix
[1 2
A
=
over R.
(c) Working over the field F3 with 3 elements, use row and column operations to put
the matrix
[6]
0123]
A
=
3210
into canonical form for equivalence and write down the canonical form. What is
the rank of A as a matrix over F3?
4
Chapter 5 Solutions
Contemporary Abstract Algebra
Ch. 5 - Let [123456213546]and=[123456612435] . Compute...Ch. 5 - Let [1234567823451786]and=[1234567813876524] ....Ch. 5 - Write each of the following permutations as a...Ch. 5 - Find the order of each of the following...Ch. 5 - What is the order of each of the following...Ch. 5 - What is the order of each of the following...Ch. 5 - What is the order of the product of a pair of...Ch. 5 - Determine whether the following permutations are...Ch. 5 - What are the possible orders for the elements of...Ch. 5 - Show that A8 contains an element of order 15.
Ch. 5 - Find an element in A12 of order 30.Ch. 5 - Show that a function from a finite set S to itself...Ch. 5 - Prob. 13ECh. 5 - Suppose that is a 6-cycle and is a 5-cycle....Ch. 5 - Prob. 15ECh. 5 - If is even, prove that 1 is even. If is odd,...Ch. 5 - Prob. 17ECh. 5 - In Sn , let be an r-cycle, an s-cycle, and a...Ch. 5 - Let and belong to Sn . Prove that is even if...Ch. 5 - Associate an even permutation with the number +1...Ch. 5 - Complete the following statement: A product of...Ch. 5 - What cycle is (a1a2an)1 ?Ch. 5 - Show that if H is a subgroup of Sn , then either...Ch. 5 - Suppose that H is a subgroup of Sn of odd order....Ch. 5 - Give two reasons why the set of odd permutations...Ch. 5 - Let and belong to Sn . Prove that 11 is an...Ch. 5 - Prob. 27ECh. 5 - How many elements of order 5 are in S7 ?Ch. 5 - Prob. 29ECh. 5 - Prove that (1234) is not the product of 3-cycles....Ch. 5 - Let S7 and suppose 4=(2143567) . Find . What are...Ch. 5 - My mind rebels at stagnation. Give me problems,...Ch. 5 - Let (a1a2a3a4)and(a5a6) be disjoint cycles in S10...Ch. 5 - If and are distinct 2-cycles, what are the...Ch. 5 - Prob. 35ECh. 5 - Let =(1,3,5,7,9,8,6)(2,4,10) . What is the...Ch. 5 - Prob. 37ECh. 5 - Let H=S5(1)=1and(3)=3 . Prove that H is a...Ch. 5 - In S4 , find a cyclic subgroup of order 4 and a...Ch. 5 - In S3 , find elements and such that...Ch. 5 - Find group elements and in S5 such that...Ch. 5 - Represent the symmetry group of an equilateral...Ch. 5 - Prove that Sn is non-Abelian for all n3 .Ch. 5 - Prove that An is non-Abelian for all n4 .Ch. 5 - For n3 , let H=bSn(1)=1 or 2 and (2)=1or2 .Prove...Ch. 5 - Show that in S7 , the equation x2=(1234) has no...Ch. 5 - If (ab) and (cd) are distinct 2-cycles in Sn ,...Ch. 5 - Let and belong to Sn . Prove that 1 and are...Ch. 5 - Viewing the members of D4 as a group of...Ch. 5 - Viewing the members of D5 as a group of...Ch. 5 - Prob. 51ECh. 5 - Prob. 52ECh. 5 - Show that A5 has 24 elements of order 5, 20...Ch. 5 - Find a cyclic subgroup of A8 that has order 4....Ch. 5 - Prob. 55ECh. 5 - Prob. 56ECh. 5 - Show that every element in An for n3 can be...Ch. 5 - Show that for n3,Z(Sn)=[] .Ch. 5 - Prob. 59ECh. 5 - Use the Verhoeff check-digit scheme based on D5 to...Ch. 5 - Prob. 61ECh. 5 - (Indiana College Mathematics Competition) A...Ch. 5 - Prob. 63ECh. 5 - Find five subgroups of S5 of order 24.Ch. 5 - Why does the fact that the orders of the elements...Ch. 5 - Let a belong to Sn . Prove that divides n!Ch. 5 - Encrypt the message ATTACK POSTPONED using the...Ch. 5 - The message VAADENWCNHREDEYA was encrypted using...
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