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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.Find all homomorphic images of the quaternion group.34. Let be the set of eight elements with identity element and noncommutative multiplication given by for all in (The circular order of multiplication is indicated by the diagram in Figure .) Given that is a group of order , write out the multiplication table for . This group is known as the quaternion group. (Sec. Sec. Sec. Sec. Sec. Sec. Sec. ) Sec. 22. Find the center for each of the following groups . a. in Exercise 34 of section 3.1. 32. Find the centralizer for each element in each of the following groups. a. The quaternion group in Exercise 34 of section 3.1 Sec. 2. Let be the quaternion group. List all cyclic subgroups of . Sec. 11. The following set of matrices , , , , , , forms a group with respect to matrix multiplication. Find an isomorphism from to the quaternion group. Sec. 8. Let be the quaternion group of units . Sec. 23. Find all subgroups of the quaternion group. Sec. 40. Find the commutator subgroup of each of the following groups. a. The quaternion group . Sec. 3. The quaternion group ; . 11. Find all homomorphic images of the quaternion group. 16. Repeat Exercise with the quaternion group , the Klein four group , and defined by
- 6. Let (image):(a) Find a basis for the null space of A.(b) Find a basis for the column space of A.(c) What are the rank and nullity of A?(d) Show that the rank-nullity theorem is verified.Question1-: (Filter generated by a family of sets). Let S ⊆ ℘(X) be nonempty family of subsets of X, and let BS be the family of all finite intersections of elements of S (note that S ⊆ BS). Show that BS is a basis for a filter. Show that this filter is nontrivial if and only if S has the finite intersection property.1 Moreover, prove that this is the smallest filter containing S. (Note that therefore there exist families that are not contained in any nontrivial filter). Question2-: Let F be a nontrivial on X. Prove that the follow- ing statements are equivalent1. F is an ultrafilter.2. (∀A⊆X)A∈F ↔ X−A∈/F.3. (∀A,B⊆X)A∪B∈F → A∈F or B∈F.Classify all Lie algebras of dimenision four and rank 1; in particular, show that they are all direct sums of Lie algebras described above
- Prove that C5 is isomorphic to its complement. Show your work and complete the proof.Let L be a finite dimensional Lie algebra. Any finite dimensional representation of L is completely reducible. Show that L must be semisimple. (converse to Weyl's Theorem)Let R be a partial order on P. (a) Prove: R-1 is a partial order on P. (b) How can you get the Hasse diagram of (P, R-1) from the Hasse diagram of (P, R)?
- which is the collection of all polynomials of degree ≤ 3. Write out the standard basis for P2? What is the dimension of P2? Is it possible for the dimension to be some other number as well? Explain. (2) Why is the following true? If {p1, p2, p3} spans P2 then it is a basis for P2. (1) Let p1 = 2−x+x2 , p2 = 1+x, p3 = x+x2 . Show that S = {p1, p2, p3} spans P2. Conclude that S is a basis for P2. (5) Using (2.3) or otherwise, write p = 3 + 5x − 4x2 as a linear combination of p1, p2 and p3. Show all working. Hence find (p)S, the coordinate vector of p relative to S. (2) Explain why are the vectors q1 = 8 + 4x − 6x2 and q2 = −4 − 2x + 3x2 are linearly dependent in P2? (2)A necessary and sufficient condition that a linear transformation P on a complex inner product space V be self adjoint., Prove the both parts , and only handwriting solution , don't copy from other sources?Let R be a ring. Show that an R-module M is isomorphic to direct sum of A1 and A2 if and only if there are R-homomorphisms f_i: A_i---->M, for i=1,2, such that for any module B and R-homomorphisms g_i:A_i----->B, for i=1,2, there is unique R-homomorphism h:M--->B so that g_i=h(f)
