The direct decompositions of a group $G$ with $G/G’$ finitely generated

Abstract:

We consider the class $\mathcal {C}$ which consists of the groups $M$ with $M/M’$ finitely generated which satisfy the maximal condition on direct factors. It is well known that any $\mathcal {C}$-group has a decomposition in finite direct product of indecomposable groups, and that two such decompositions are not necessarily equivalent up to isomorphism, even for a finitely generated nilpotent group. Here, we show that any $\mathcal {C}$-group has only finitely many nonequivalent decompositions. In order to prove this result, we introduce, for $\mathcal {C}$-groups, a slightly different notion of decomposition, that we call $J$-decomposition; we show that this decomposition is necessarily unique. We also obtain, as consequences of the properties of $J$-decompositions, several generalizations of results of R. Hirshon. For instance, we have $\mathbb {Z} \times G \cong \mathbb {Z} \times H$ for any groups $G$, $H$ which satisfy $M \times G \cong M \times H$ for a $\mathcal {C}$-group $M$.