A subgroup of a group must be closed under the same operation of the group (here matrix multiplication). In other words, for two Lorentz transformations and from a particular set, the composite Lorentz transformations and must be in the same set as and . This is not always the case: the composition of two antichronous Lorentz transformations is orthochronous, and the composition of two improper Lorentz transformations is proper. In other words, while the sets formula_37, formula_38, formula_39, and formula_40 all form subgroups, the sets containing improper and/or antichronous transformations without enough proper orthochronous transformations (e.g. formula_41, formula_42, formula_43) do not form subgroups.