A group \(G\) is defined to have property \({\mathcal P}_ 1\) if there exists a \({\mathbb{Z}}\)-torsionfree \({\mathbb{Z}}G\)-module \(T\) with \(\text{proj.}\dim T \leq 1\) and \(H^ 0 (G, T) \neq 0\). Clearly finite groups have property \({\mathcal P}_ 1\). More interestingly a group \(G\) with period \(q\) after 1 step (i.e. there exists \(q > 1\) such that \(H^ i (G, -)\) and \(H^{i + q} (G, -) \) are naturally equivalent for all \(i > 1\)) has property \({\mathcal P}_ 1\). The author shows that if \(G\) is a countable infinite group with property \({\mathcal P}_ 1\), then \(H^ 1(G,\mathbb{Z} G) \neq 0\), and thus, by Stallings’ structure theorem, \(G\) is (modulo accessibility) the fundamental group of a graph of finite groups. This eventually leads to the result that a torsion group \(G\) has property \({\mathcal P}_ 1\) if and only if it is a countable locally finite group.