1. Prove or disprove each of the following statements.🔗 All of the generators of Z60 are prime.🔗 🔗 U(8) is cyclic.🔗 🔗 Q is cyclic.🔗 🔗 If every proper subgroup of a group G is cyclic, then G is a cyclic group.🔗 🔗 A group with a finite number of subgroups is finite.🔗 🔗 🔗
2. Find the order of each of the following elements.🔗 5∈Z12🔗 🔗 3∈R🔗 🔗 3∈R∗🔗 🔗 −i∈C∗🔗 🔗 72 in Z240🔗 🔗 312 in Z471🔗 🔗 🔗
3. List all of the elements in each of the following subgroups.🔗 The subgroup of Z generated by 7🔗 🔗 The subgroup of Z24 generated by 15🔗 🔗 All subgroups of Z12🔗 🔗 All subgroups of Z60🔗 🔗 All subgroups of Z13🔗 🔗 All subgroups of Z48🔗 🔗 The subgroup generated by 3 in U(20)🔗 🔗 The subgroup generated by 5 in U(18)🔗 🔗 The subgroup of R∗ generated by 7🔗 🔗 The subgroup of C∗ generated by i where i2=−1🔗 🔗 The subgroup of C∗ generated by 2i🔗 🔗 The subgroup of C∗ generated by (1+i)/2🔗 🔗 The subgroup of C∗ generated by (1+3i)/2🔗 🔗 🔗
4. Find the subgroups of GL2(R) generated by each of the following matrices.🔗 (01−10)🔗 🔗 (01/330)🔗 🔗 (1−110)🔗 🔗 (1−101)🔗 🔗 (1−1−10)🔗 🔗 (3/21/2−1/23/2)🔗 🔗 🔗
10. Find all elements of finite order in each of the following groups. Here the “∗” indicates the set with zero removed.🔗 Z🔗 🔗 Q∗🔗 🔗 R∗🔗 🔗 🔗
12. Find a cyclic group with exactly one generator. Can you find cyclic groups with exactly two generators? Four generators? How about n generators?🔗 🔗
13. For ,n≤20, which groups U(n) are cyclic? Make a conjecture as to what is true in general. Can you prove your conjecture?🔗 🔗
14. Let andA=(01−10)andB=(0−11−1) be elements in .GL2(R). Show that A and B have finite orders but AB does not. 🔗 🔗
15. Evaluate each of the following.🔗 (3−2i)+(5i−6)🔗 🔗 (4−5i)−(4i−4)―🔗 🔗 (5−4i)(7+2i)🔗 🔗 (9−i)(9−i)―🔗 🔗 i45🔗 🔗 (1+i)+(1+i)―🔗 🔗 🔗
16. Convert the following complex numbers to the form .a+bi.🔗 2cis(π/6)🔗 🔗 5cis(9π/4)🔗 🔗 3cis(π)🔗 🔗 cis(7π/4)/2🔗 🔗 🔗
17. Change the following complex numbers to polar representation.🔗 1−i🔗 🔗 −5🔗 🔗 2+2i🔗 🔗 3+i🔗 🔗 −3i🔗 🔗 2i+23🔗 🔗 🔗
18. Calculate each of the following expressions.🔗 (1+i)−1🔗 🔗 (1−i)6🔗 🔗 (3+i)5🔗 🔗 (−i)10🔗 🔗 ((1−i)/2)4🔗 🔗 (−2−2i)12🔗 🔗 (−2+2i)−5🔗 🔗 🔗
19. Prove each of the following statements.🔗 |z|=|z―|🔗 🔗 zz―=|z|2🔗 🔗 z−1=z―/|z|2🔗 🔗 |z+w|≤|z|+|w|🔗 🔗 |z−w|≥||z|−|w||🔗 🔗 |zw|=|z||w|🔗 🔗 🔗
20. List and graph the 6th roots of unity. What are the generators of this group? What are the primitive 6th roots of unity?🔗 🔗
21. List and graph the 5th roots of unity. What are the generators of this group? What are the primitive 5th roots of unity?🔗 🔗
22. Calculate each of the following.🔗 2923171(mod582)🔗 🔗 2557341(mod5681)🔗 🔗 20719521(mod4724)🔗 🔗 971321(mod765)🔗 🔗 🔗
23. Let .a,b∈G. Prove the following statements.🔗 The order of a is the same as the order of .a−1.🔗 🔗 For all ,g∈G, .|a|=|g−1ag|.🔗 🔗 The order of ab is the same as the order of .ba.🔗 🔗 🔗
30. Suppose that G is a group and let ,a, .b∈G. Prove that if |a|=m and |b|=n with ,gcd(m,n)=1, then .⟨a⟩∩⟨b⟩={e}.🔗 🔗
31. Let G be an abelian group. Show that the elements of finite order in G form a subgroup. This subgroup is called the torsion subgroup of .G.🔗 🔗
32. Let G be a finite cyclic group of order n generated by .x. Show that if y=xk where ,gcd(k,n)=1, then y must be a generator of .G.🔗 🔗
33. If G is an abelian group that contains a pair of cyclic subgroups of order 2, show that G must contain a subgroup of order 4. Does this subgroup have to be cyclic?🔗 🔗
34. Let G be an abelian group of order pq where .gcd(p,q)=1. If G contains elements a and b of order p and q respectively, then show that G is cyclic.🔗 🔗
39. Prove that if G is a cyclic group of order m and ,d∣m, then G must have a subgroup of order .d.🔗 🔗
41. If z=r(cosθ+isinθ) and w=s(cosϕ+isinϕ) are two nonzero complex numbers, show that zw=rs[cos(θ+ϕ)+isin(θ+ϕ)]. 🔗 🔗