(a)Let Re denote the rotation matrixcos esin 6sin eRecos 6Show that the set of Ro. where @ varies through the real num-bers, forms a group under matrix multiplication. In particular,Re RLet J denote the matrix of reflection in the x-axis,Ras and R, = Rg.(b)Show /Ra RJ

(a) To prove: The set {Rθ=cosθ-sinθsinθcosθ | θ∈ℝ} forms a group under matrix multiplication. (b)…

Answered: (a) Let Re denote the rotation matrix…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.2: Properties Of Group Elements

Problem 3E: Prove part e of Theorem 3.4. Theorem 3.4: Properties of Group Elements Let G be a group with respect...

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Find the centralizer for each element a in each of the following groups. The quaternion group G={ 1,i,j,k,1,i,j,k } in Exercise 34 of section 3.1 (Sec. 3.1, #34). G={ I2,R,R2,R3,H,D,V,T } in Exercise 36 of section 3.1 (Sec. 3.1, #36). G={ I3,P1,P2,P3,P4,P5 } in Exercise 35 of section 3.1 (Sec. 3.1, #35). Sec. 3.1,34 34. Let G be the set of eight elements G={ 1,i,j,k,1,i,j,k } with identity element 1 and noncommutative multiplication given by (1)2=1, i2=j2=k2=1, ij=ji=k jk=kj=i, ki=ik=j, x=(1)x=x(1) for all x in G (The circular order of multiplication is indicated by the diagram in Figure 3.8.) Given that G is a group of order 8, write out the multiplication table for G. This group is known as the quaternion group. (Sec. 3.3,22a,32a, Sec. 3.4,2, Sec. 3.5,11, Sec. 4.2,8, Sec. 4.4,23, Sec. 4.5,40a, Sec. 4.6,3,11,16) Sec. 3.1,36 Consider the matrices R=[ 0110 ] H=[ 1001 ] V=[ 1001 ] D=[ 0110 ] T=[ 0110 ] in GL(2,), and let G={ I2,R,R2,R3,H,D,V,T }. Given that G is a group of order 8 with respect to multiplication, write out a multiplication table for G. (Sec. 3.3,22b,32b, Sec. 4.1,22, Sec. 4.6,14) Sec. 3.1,35 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.
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.
11. Show that is a generating set for the additive abelian group if and only if
Let G=I2,R,R2,R3,H,D,V,T be the multiplicative group of matrices in Exercise 36 of Section 3.1, let G=1,1 under multiplication, and define :GG by ([ abcd ])=adbc a. Assume that is an epimorphism, and find the elements of K=ker. b. Write out the distinct elements of G/K. c. Let :G/KG be the isomorphism described in the proof of Theorem 4.27, and write out the values of . Consider the matrices R=[ 0110 ] H=[ 1001 ] V=[ 1001 ] D=[ 0110 ] T=[ 0110 ] in GL(2,), and let G={ I2,R,R2,R3,H,D,V,T }. Given that G is a group of order 8 with respect to multiplication, write out a multiplication table for G.
Rework exercise 9 with G=GL(2,), the general linear group of order 2 over , and G= under addition. M2() Exercise 9: Let G be the additive group of 22 matrices over and G the additive group of real numbers. Define :GG by ([ abcd ])=a+d. (This mapping is called the trace of the matrix.) Prove or disprove that is a homomorphism. If is a homomorphism, find ker and decide whether is an epimorphism or a monomorphism.
23. Let be a group that has even order. Prove that there exists at least one element such that and . (Sec. ) Sec. 4.4, #30: 30. Let be an abelian group of order , where is odd. Use Lagrange’s Theorem to prove that contains exactly one element of order .
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.
11. Let be , and let be the group of nonzero real numbers under multiplication. Prove that the mapping defined by Is a homomorphism, and find ker . Is an epimorphism? Is a monomorphism? (The value of this mapping is called the determinant of the matrix.)
True or False Label each of the following statements as either true or false. 11. The invertible elements of form an abelian group with respect to matrix multiplication.
Let G=1,i,j,k be the quaternion group. List all cyclic subgroups of G. (Sec. 3.1,34).
2. Show that is a normal subgroup of the multiplicative group of invertible matrices in .
Let a and b be elements of a group G. Prove that G is abelian if and only if (ab)2=a2b2.

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(a) Let Re denote the rotation matrix cos e Re sin@ sin Cos 0 Show that the set of Rg. where 0 varies through the real num- bers, forms a group under matrix multiplication. In particular, Re R Let J denote the matrix of reflection in the x-axis, Re+ and R = R g. (b) Show JRo = RJ