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 Λ and L from a particular subgroup, the composite Lorentz transformations Λ L and L Λ must be in the same subgroup as Λ and L. 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 L + ↑ ↑ , L + , L ↑ ↑ , and L 0 = L + ↑ ↑ ∪ ∪ L − − ↓ ↓ all form subgroups, the sets containing improper and/or antichronous transformations without enough proper orthochronous transformations (e.g. L + ↓ ↓ , L − − ↓ ↓ , L − − ↑ ↑ ) do not form subgroups.