By modifying the firing rule of Petri nets, we define strict-Z-nets and we generalize the latter to strict-G-nets for an arbitrary group G. We show that strict-Z-nets are an extension of pure Petri nets and we classify strict-G-nets for various groups G with respect to usual classes of Petri nets. In order to cover both strict-Z-nets and general Petri nets, we add to our nets new arcs with relaxed firing conditions, thus obtaining Z-nets, and more generally group-nets. We conclude by comparing group-nets with Petri nets.