We will begin by presenting some history of the Krull-Schmidt-Remak Theorem. From groups, we will pass to modules over a ring R, introducing some direct-sum decompositions that follow a special pattern. We will consider invariants that also appear in factorisation of polynomials. Then we will go back from (right) R-modules to groups. Here the category that appears in a natural way is that of G-groups, which substitutes the category of right R-modules. In this category, Remak's result has a natural interpretation.