Introduction to Abstract Algebra, Solutions Manual, 4th Edition

2.4 Cyclic Groups and the Order of an Element

1. If o(g) = n, we use Theorem 8: g^k generates G = g if and only if gcd (k, n) = 1.

a. o(g) = 5. Then G = g^k if k = 1, 2, 3, 4.

c. o(g) = 16. Then G = g^k if k = 1, 3, 5, 7, 9, 11, 13, 15.

2. Since is cyclic and , the solution to Exercise 1 applies.

a. has generators .

2. has generators

3. a. G = g , o(g) =∞. We claim g and g⁻¹ are the only ...