A group is said to be indecomposable if it cannot be written as a product of smaller groups. An example of this is any group of prime order p, which is isomorphic to the group of integers modulo p (with addition as the group operation). Since the integers modulo p is a cyclic group (generated by 1), we have that any indecomposable group of prime order is cyclic. I have two questions: Are ALL indecomposable groups cyclic? (N.B. not just those of prime order) Are all cyclic groups indecomposable? Thanks!