b. Show that a cyclic group of order 6 has a cyclic group of order 2 as a homomorphic image.

27. a. Show that a cyclic group of order has a cyclic group of order as a homomorphic image. b. Show that a cyclic group of order has a cyclic group of order as a homomorphic image.

If G is a cyclic group, prove that the equation x2=e has at most two distinct solutions in G.

9. Find all homomorphic images of the octic group.

Let G be an abelian group of order 2n, where n is odd. Use Lagranges Theorem to prove that G contains exactly one element of order 2.

9. Suppose that and are subgroups of the abelian group such that . Prove that .

Exercises 35. Prove that any two groups of order are isomorphic.

Let be a group of order , where and are distinct prime integers. If has only one subgroup of order and only one subgroup of order , prove that is cyclic.

Prove or disprove that H={ hGh1=h } is a subgroup of the group G if G is abelian.

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)