Lattice of subgroups
In mathematics, the lattice of subgroups of a group formula_1 is the lattice whose elements are the subgroups of formula_1, with the partial order relation being set inclusion. In this lattice, the join of two subgroups is the subgroup generated by their union, and the meet of two subgroups is their intersection. Example. The dihedral group Dih has ten subgroups, counting itself and the trivial subgroup. Five of the eight group elements generate subgroups of order two, and the other two non-identity elements both generate the same cyclic subgroup of order four. In addition, there are  two subgroups of the form Z × Z, generated by pairs of order-two elements. The lattice formed by these ten subgroups is shown in the illustration. This example also shows that the lattice of all subgroups of a group is not a modular lattice in general. Indeed, this particular lattice contains the forbidden "pentagon" "N" as a sublattice. Properties. For any "A", "B", and "C" subgroups of a group with "A" ≤ "C" ("A" subgroup of "C") then "AB" ∩ "C" = "A(B ∩ C)"; the multiplication here is the product of subgroups. This property has been called the  modular property of groups or "(Dedekind's) modular law" (, ). Since for two normal subgroups the product is actually the smallest subgroup containing the two, the normal subgroups form a modular lattice. The Lattice theorem establishes a Galois connection between the lattice of subgroups of a group and that of its quotients. The Zassenhaus lemma gives an isomorphism between certain combinations of quotients and products in the lattice of subgroups. In general, there is no restriction on the shape of the lattice of subgroups, in the sense that every lattice is isomorphic to a sublattice of the subgroup lattice of some  group. Furthermore, every finite lattice is isomorphic to a sublattice of the subgroup lattice of some finite group . Characteristic lattices. Subgroups with certain properties form lattices, but other properties do not. However, neither finite subgroups nor torsion subgroups form a lattice: for instance, the free product formula_3 is generated by two torsion elements, but is infinite and contains elements of infinite order. The fact that normal subgroups form a modular lattice is a particular case of a more general result, namely that in any Maltsev variety (of which groups are an example), the lattice of congruences is modular . Characterizing groups  by their subgroup lattices. Lattice theoretic information about the lattice of subgroups can sometimes be used to infer information about the original group, an idea that goes back to the work of . For instance, as Ore proved, a group is locally cyclic if and only if its lattice of subgroups is distributive. If additionally the lattice satisfies the ascending chain condition, then the group is cyclic. The groups whose lattice of subgroups is a complemented lattice are called complemented groups , and the groups whose lattice of subgroups are modular lattices are called Iwasawa groups or modular groups . Lattice-theoretic characterizations  of this type also exist for solvable groups and perfect groups .