Exercise 2.2.6 Consider the affine groupAM9) = { (; ")-bez, a#0}.Show that Aff(3) is a group under matrix multiplication as defined in formula (1.3) ofSection 1.8 and the statement that follows it. Draw a Cayley graph for this group with{(6 ):6 1)} Compare with the Cayley graph X (Da, {R, F} ) Aregenerating set S=these groups really the same in some sense (a sense to be known to us as isomorphic groupsin Section 3.2)?

Given: Let us consider the given affine group Aff3ab01 a,b∈ℤ3,a≠0 Draw a Cayley graph for this…

Answered: Exercise 2.2.6 Consider the affine…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.4: Cyclic Groups

Problem 11E: Exercises 11. According to Exercise of section, if is prime, the nonzero elements of form a...

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31. (See Exercise 30.) Prove that if and are primes and is a nonabelian group of order , then the center of is the trivial subgroup . Exercise 30: 30. Let be a group with center . Prove that if is cyclic, then is abelian.
5. Exercise of section shows that is a group under multiplication. a. List the elements of the subgroupof , and state its order. b. List the elements of the subgroupof , and state its order. Exercise 33 of section 3.1. a. Let . Show that is a group with respect to multiplication in if and only if is a prime. State the order of . This group is called the group of units in and is designated by . b. Construct a multiplication table for the group of all nonzero elements in , and identify the inverse of each element.
Find two groups of order 6 that are not isomorphic.
Find all subgroups of the quaternion group.
2. Show that is a normal subgroup of the multiplicative group of invertible matrices in .
Find all homomorphic images of the quaternion group.
9. Suppose that and are subgroups of the abelian group such that . Prove that .
15. 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.
4. Prove that the special linear group is a normal subgroup of the general linear group .
Prove that the Cartesian product 24 is an abelian group with respect to the binary operation of addition as defined in Example 11. (Sec. 3.4,27b, Sec. 5.1,53,) Example 11. Consider the additive groups 2 and 4. To avoid any unnecessary confusion we write [ a ]2 and [ a ]4 to designate elements in 2 and 4, respectively. The Cartesian product of 2 and 4 can be expressed as 24={ ([ a ]2,[ b ]4)[ a ]22,[ b ]44 } Sec. 3.4,27b 27. Prove or disprove that each of the following groups with addition as defined in Exercises 52 of section 3.1 is cyclic. a. 23 b. 24 Sec. 5.1,53 53. Rework Exercise 52 with the direct sum 24.
Prove or disprove that the set of all diagonal matrices in Mn() forms a group with respect to addition.
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.

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