More discrete copies of Z in βN

In a previous paper we established that if q is any minimal idempotent in beta N, then for all except possibly one p is an element of cl {2(n): n is an element of N} \ N, q + p + q generates an infinite discrete group. Responding to a question of Wis Comfort, we extend this result in two directions. We show on the one hand that for a minimal idempotent q, there is at most one prime r for which there exists p is an element of cl {r(n): n is an element of N} \ N such that the group generated by q + p + q is not both infinite and discrete. On the other hand, we show that for any p is an element of beta N, if p is an element of cl (nN) for infinitely many n is an element of N, then there is some minimal idempotent q such that the group generated by q + p + q is infinite and discrete. We also show that if G is a countable discrete group and if p is a right cancelable element of G*, then there is an idempotent q is an element of G* such that q . p . q generates a discrete copy of Z in G*. We do not know whether there exist any minimal idempotent q and any p with p is an element of cl (nN) for infinitely many n is an element of E N such that the group generated by q + p + q is not discrete. We show that if such a "bad" q exists, then there are many of them. (C) 2010 Elsevier BM. All rights reserved.