I'll start by giving a brief survey of some of the more well-known examples of pathological behaviour related to direct sum decompositions. Then I'll answer a less well-known question: is there an abelian group G that is not isomorphic to G+Z (its direct sum with an infinite cyclic group) but is isomorphic to G+Z+Z? It turns out that this was answered by Eklof and Shelah in the 1980s, but the example I'll give is significantly simpler. Also, the same group gives the simplest example I know of several other kinds of pathological behaviour.

 

This will be a low tech talk, and almost nothing I say will require more than undergraduate algebra.

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