Question 1. Let G = Z₂0 and H =< 5 >, the subgroup of Z20 generated by 5. 1. What is |G:H|? 2. Find all the left costs of H in G

22. Find the center for each of the following groups . a. in Exercise 34 of section 3.1. b. in Exercise 36 of section 3.1. c. in Exercise 35 of section 3.1. d., the general linear group of order over. Exercise 34 of section 3.1. 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. Exercise 36 of section 3.1 Consider the matrices in , and let . Given that is a group of order 8 with respect to multiplication, write out a multiplication table for. Exercise 35 of section 3.1. A permutation matrix is a matrix that can be obtained from an identity matrix by interchanging the rows one or more times (that is, by permuting the rows). For the permutation matrices are and the five matrices. Given that is a group of order with respect to matrix multiplication, write out a multiplication table for .

Find the normalizer of the subgroup (1),(1,3)(2,4) of the octic group D4.

Question 4 Let B = {a, b, c} and S is the relation on B such that S%3D ((а, a), (а, b), (b, c), (а, с), (с, с)}. Determine if S is an antisymmetric relation on B.