The Category of Formations of Finite Groups and Topology

We study classes of finite groups and their closure operators from a categorical and topological point of view. We show that the Gaschütz product, which is associative and unital on formations, is a bifunctor on the category of subgroup-closed formations, making the latter a monoidal category, and that the closure operators on the universe of finite groups form a complete lattice CL whose joins can be described through closed classes. We then use the correspondences of Kuratowski and Alexandrov between additive closure operators and topologies. The operators s, sn and Q induce pairwise distinct Alexandrov topologies which are all contractible, because the trivial group is a least element of each of the associated preorders; by contrast, EΦ and Ep induce disconnected spaces with contractible components. Finally, neither the formation operator Q∨R0 nor the Fitting class operator sn∨N0 is finitely additive, so neither of them is the closure operator of a topology on the universe of finite groups; in both cases, the obstruction comes from the synthesis operator, R0, respectively, N0.