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Concept
Lattice of subgroups
Summary
In mathematics, the lattice of subgroups of a group is the lattice whose elements are the subgroups of , 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.
The dihedral group Dih4 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 Z2 × Z2, 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" N5 as a sublattice.
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 .
Subgroups with certain properties form lattices, but other properties do not.
Normal subgroups always form a modular lattice. In fact, the essential property that guarantees that the lattice is modular is that subgroups commute with each other, i.
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