In Exercises 7 and 8, let
Let
Find the distinct left cosets of
Find the distinct right cosets of
Want to see the full answer?
Check out a sample textbook solutionChapter 4 Solutions
ELEMENTS OF MODERN ALGEBRA
- In Exercises 1- 9, let be the given group. Write out the elements of a group of permutations that is isomorphic to, and exhibit an isomorphism from to this group. 6. Let be the group of permutations matrices as given in Exercise 35 of section 3.1. Sec A permutation matrix is a matrix that can be obtained from an identity matrix by interchanging the rows one or more times (that is, by permuting the rows). For the permutation matrices are and the five matrices. Given that is a group of order with respect to matrix multiplication, write out a multiplication table for .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_forwardFind the normalizer of the subgroup (1),(1,3)(2,4) of the octic group D4.arrow_forward
- 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_forwardFind two groups of order 6 that are not isomorphic.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
- 12. 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_forwardWrite 20 as the direct sum of two of its nontrivial subgroups.arrow_forward11. Find all normal subgroups of the alternating group .arrow_forward
- 21. Let Be the special linear group of order over .Find the inverse of each of the following elements. a. b. c. d.arrow_forwardThe gray square in Table 1 has the following vertices: [00],[10],[11],[01] Apply each of the three transformations given in Table 1 to these vertices and sketch the result to verify that each transformation has the indicated effect. Use c=2 in the expansion matrix and c=1 in the shear matrix.arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage LearningAlgebra and Trigonometry (MindTap Course List)AlgebraISBN:9781305071742Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage Learning
- College AlgebraAlgebraISBN:9781305115545Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage Learning