The subset D = {(m, n) | m, ne Z and 2m+ n = 0} of the group Z x ZZ under addition

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.

9. Find all homomorphic images of the octic group.

Prove statement d of Theorem 3.9: If G is abelian, (xy)n=xnyn for all integers n.