A fine structure theorem for certain \(O\)-minimal groups is given. These results can be considered as \(O\)-minimal analogues of various results on locally modular weakly minimal groups. The authors define the property ``CF'' (collapse of functions) which is a kind of 1-basedness, as follows. The \(O\)-minimal structure \((M,<,\dots)\) has the CF property if whenever \(\{f_ u: u\in U\}\) is a uniformly definable family of partial functions from \(M\) to \(M\), and \(a\in M\), then the family of germs of the \(f_ u\) at the point \(a\), has dimension at most 1. \(O\)-minimal structures of the form \((M,+,<,\dots)\) which have the CF property are studied, where \(+\) is either (i) a (continuous) commutative group operation, or a (ii) continuous local commutative group operation. It is first shown that all definable structure on \(M\) comes from definable partial endomorphisms. It is then shown that in case (i), \(M\) is the reduct of an ordered vector space over an ordered division ring, and that in case (ii), \(M\) is the reduct of an interval in such an ordered vector space.