(a) Find Z(H). (b) Prove that Z(H)=Q, Q is under addition.
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- True or False Label each of the following statements as either true or false. 10. The nonzero elements of form a group with respect to matrix multiplication.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.2. Show that is a normal subgroup of the multiplicative group of invertible matrices in .
- True or False Label each of the following statements as either true or false. 9. The nonzero elements of form a group with respect to matrix multiplication.Let a and b be elements of a group G. Prove that G is abelian if and only if (ab)2=a2b2.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.Which of the following subsets in the set R^2×2 is not a group under the usual matrix multiplication operation? Solve (d) and (e)Let S= \ {-1} and define an operation on S by a*b = a + b + ab. Prove that (S,*) is an abelian group.
- (1) Show H(R) is a group under matrix multiplication, called the Heisenberg Group.(2) Find an explicit example of matrices A, B ∈ H(R) such that AB ̸= BA.(3) Is H(R) a subset of GL3 (R)?Which of the following pairs (S, °) forms a group? (a) S = {all surjections Z2 -> Z2}, ° is function composition. (b) S = {all injections Z -> Z}. ° is function composition. (c) S = N, ° is the usual addition of natural numbers. (d) S = R, ° is the usual multiplication of real numbers. (e) S= {[a b; c d] where a,b,c ∈ R.}. ° is the usual matrix multiplication.In the given question find an elementary matrix E such that B=EA.