Just 4e Let T be the group of nonsingular upper triangular 2 × 2 matrices with entries in R;that is, matrices of the form4.(: )where a, b, c E R and ac # 0. Let U consist of matrices of the form(6 ).where a € R.(a) Show that U is a subgroup of T.(b) Prove that U is abelian.(c) Prove that U is normal in T.(d) Show that T/U is abelian.(e) Is T normal in GL2(R)?

Answered: Image /qna-images/answer/644ef7db-0b49-491d-ae27-648eaff3e6d1.jpg

Answered: Let T be the group of nonsingular upper…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.5: Isomorphisms

Problem 5E

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14. Let be an abelian group of order where and are relatively prime. If and , 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.
Let G be a group with center Z(G)=C. Prove that if G/C is cyclic, then G is abelian.
Let T be a linear transformation T such that T(v)=kv for v in Rn. Find the standard matrix for T.
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.
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.
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 .
Complete Example 2 by verifying that {1,x,x2,x3} is an orthonormal basis for P3 with the inner product p,q=a0b0+a1b1+a2b2+a3b3. An Orthonormal basis for P3. In P3, with the inner product p,q=a0b0+a1b1+a2b2+a3b3 The standard basis B={1,x,x2,x3} is orthonormal. The verification of this is left as an exercise See Exercise 17..
Let G be an abelian group of order 2n, where n is odd. Use Lagranges Theorem to prove that G contains exactly one element of order 2.
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.
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.)

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