In Exercises 1- 9, let
Let
Sec
A permutation matrix is a matrix that can be obtained from an identity matrix
Given that
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Chapter 4 Solutions
Elements Of Modern Algebra
- The elements of the multiplicative group G of 33 permutation matrices are given in Exercise 35 of section 3.1. Find the order of each element of the group. (Sec. 3.1,35) A permutation matrix is a matrix that can be obtained from an identity matrix In by interchanging the rows one or more times (that is, by permuting the rows). For n=3 the permutation matrices are I3 and the five matrices. (Sec. 3.3,22c,32c, Sec. 3.4,5, Sec. 4.2,6) P1=[ 100001010 ] P2=[ 010100001 ] P3=[ 010001100 ] P4=[ 001010100 ] P5=[ 001100010 ] Given that G={ I3,P1,P2,P3,P4,P5 } is a group of order 6 with respect to matrix multiplication, write out a multiplication table for G.arrow_forwardProve or disprove that the set of all diagonal matrices in Mn() forms a group with respect to addition.arrow_forward12. Find all homomorphic images of each group in Exercise of Section. 18. Let be the group of units as described in Exercise. For each value of, write out the elements of and construct a multiplication table for . a. b. c. d.arrow_forward
- Find two groups of order 6 that are not isomorphic.arrow_forward15. Repeat Exercise with, the multiplicative group of matrices in Exercise of Section. 14. Let be the multiplicative group of matrices in Exercise of Section, let under multiplication, and define by a. Assume that is an epimorphism, and find the elements of. b. Write out the distinct elements of. c. Let be the isomorphism described in the proof of Theorem, and write out the values of.arrow_forwardExercises 1. Express each permutation as a product of disjoint cycles and find the orbits of each permutation. a. b. c. d. e. f. g. h.arrow_forward
- In Exercises 7 and 8, let be the multiplicative group of permutation matrices in Example 6 of Section 3.5 Let be the subgroup of given by . Find the distinct left cosets of in , write out their elements, partition into left cosets of , and give . Find the distinct right cosets of in , write out their elements, and partition into right cosets of .arrow_forwardFind the order of each permutation in Exercise 1. Express each permutation as a product of disjoint cycles and find the orbits of each permutation. a. [ 1234545312 ] b. [ 1234513254 ] c. [ 1234541352 ] d. [ 1234535241 ] e. [ 12345673456127 ] f. [ 12345675137264 ] g. [ 1234513452 ][ 1234532415 ] h. [ 1234523415 ][ 1234513542 ]arrow_forward38. Let be the set of all matrices in that have the form with all three numbers , , and nonzero. Prove or disprove that is a group with respect to multiplication.arrow_forward
- 40. Prove or disprove that the set in Exercise is a group with respect to addition. 38. Let be the set of all matrices in that have the form with all three numbers , , and nonzero. Prove or disprove that is a group with respect to multiplication.arrow_forwardShow that a group of order 4 either is cyclic or is isomorphic to the Klein four group e,a,b,ab=ba.arrow_forwardFind the normalizer of the subgroup (1),(1,3)(2,4) of the octic group D4.arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning