1.) Let G₁ = £1₁ 1₁ i -i) be group of 4 Complex numbers under mutiplication. a) is £1, i} a subgroup 6 // why? b.) is £1,1} a subgroup for and why?
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- 9. Find all homomorphic images of the octic group.5. Exercise of section shows that is a group under multiplication. a. List the elements of the subgroupof , and state its order. b. List the elements of the subgroupof , and state its order. Exercise 33 of section 3.1. a. Let . Show that is a group with respect to multiplication in if and only if is a prime. State the order of . This group is called the group of units in and is designated by . b. Construct a multiplication table for the group of all nonzero elements in , and identify the inverse of each element.12. Find all normal subgroups of the quaternion group.
- Let A={ a,b,c }. Prove or disprove that P(A) is a group with respect to the operation of union. (Sec. 1.1,7c)Exercises 11. According to Exercise of section, if is prime, the nonzero elements of form a group with respect to multiplication. For each of the following values of , show that this group is cyclic. (Sec. ) a. b. c. d. e. f. 33. a. Let . Show that is a group with respect to multiplication in if and only if is a prime. State the order of . This group is called the group of units in and designated by . b. Construct a multiplication table for the group of all nonzero elements in , and identify the inverse of each element.Exercises 38. Assume that is a cyclic group of order. Prove that if divides , then has a subgroup of order.
- Prove that Ca=Ca1, where Ca is the centralizer of a in the group G.12. Find all homomorphic images of each group in Exercise of Section. 18. Let be the group of units as described in Exercise. For each value of, write out the elements of and construct a multiplication table for . a. b. c. d.1. Consider , the groups of units in under multiplication. For each of the following subgroups in , partition into left cosets of , and state the index of in a. b.