Abstract Algebra Theory and Applications - Computer Science ...

EXERCISES 67The method of repeated squares will prove to be a very useful tool whenwe explore RSA cryptography in Chapter 6. To encode and decode messagesin a reasonable manner under this scheme, it is necessary to be able toquickly compute large powers of integers mod n.Exercises1. Prove or disprove each of the following statements.(a) U(8) is cyclic.(b) All of the generators of Z 60 are prime.(c) Q is cyclic.(d) If every subgroup of a group G is cyclic, then G is a cyclic group.(e) A group with a finite number of subgroups is finite.2. Find the order of each of the following elements.(a) 5 ∈ Z 12(c) √ 3 ∈ R ∗(e) 72 in Z 240(b) √ 3 ∈ R(d) −i ∈ C ∗(f) 312 in Z 4713. List all of the elements in each of the following subgroups.(a) The subgroup of Z generated by 7(b) The subgroup of Z 24 generated by 15(c) All subgroups of Z 12(d) All subgroups of Z 60(e) All subgroups of Z 13(f) All subgroups of Z 48(g) The subgroup generated by 3 in U(20)(h) The subgroup generated by 6 in U(18)(i) The subgroup of R ∗ generated by 7(j) The subgroup of C ∗ generated by i where i 2 = −1(k) The subgroup of C ∗ generated by 2i(l) The subgroup of C ∗ generated by (1 + i)/ √ 2(m) The subgroup of C ∗ generated by (1 + √ 3 i)/2