Lattice of subgroups

Hasse diagram of the l.o.s. of the dihedral group Dih4, with the subgroups represented by their cycle graphs

In mathematics, the lattice of subgroups of a group ${\displaystyle G}$ is the lattice whose elements are the subgroups of ${\displaystyle G}$, 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.

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 Template:Harvs. For instance, as Ore proved, a group is locally cyclic if and only if its lattice of subgroups is distributive. Lattice-theoretic characterizations of this type also exist for solvable groups and perfect groups Template:Harv.

Example

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 two others generate the same cyclic group C4. In addition, there are two groups of the form C2×C2, 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.

Characteristic lattices

Subgroups with certain properties form lattices, but other properties do not.

• Nilpotent normal subgroups form a lattice, which is (part of) the content of Fitting's theorem.
• In general, for any Fitting class F, both the subnormal F-subgroups and the normal F-subgroups form lattices. This includes the above with F the class of nilpotent groups, as well as other examples such as F the class of solvable groups. A class of groups is called a Fitting class if it is closed under isomorphism, subnormal subgroups, and products of subnormal subgroups.
• Central subgroups form a lattice.

However, neither finite subgroups nor torsion subgroups form a lattice: for instance, the free product ${\displaystyle \mathbf {Z} /2\mathbf {Z} *\mathbf {Z} /2\mathbf {Z} }$ is generated by two torsion elements, but is infinite and contains elements of infinite order.

References

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|CitationClass=book }}. Review by Ralph Freese in Bull. AMS 33 (4): 487–492.

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