The only examples I have encountered of infinite $p$-groups with trivial center employ non-elementary methods in their construction. For instance, Example 9.2.5 of Scott's *Group Theory* is a perfectly satisfactory example, but it requires the wreath product (which, though an invaluable group-theoretic tool, is not what I consider an "elementary method").

Does anyone know of an example (of an infinite $p$-group with trivial center) that can be constructed and proven to have the claimed properties in a way that is friendly to, say, students of a first course in group theory? Perhaps a large product of finite groups or an easy-to-describe matrix group?

(I also welcome arguments for the nonexistence of such an example!)

Every $p$-group has nontrivial centeris false for infinite $p$-groups. Other natural generalizations in basic group theory have accessible counterexamples. The one that comes to mind isA group in which every nontrivial element has order 2 is abelian. Changing '2' to '3' gives a false statement, and there is a not-too-hard-to-describe group of order 27 that illustrates this. – Zach N Dec 14 '10 at 4:53