A profinite group is said to be small if it has only finitely many open subgroups of every given index. We prove that every small profinite group can be decomposed into a direct product of indecomposable profinite groups, and that such a decomposition is unique up to order and isomorphisms of the factors.
This is closely related to the property of being cancellable in direct products. In this talk I will discuss our investigations of these properties beyond the class of small profinite groups leading to some open questions.
