Lattice of subgroups

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In mathematics, the lattice of subgroups of a group $G$ is the lattice whose elements are the subgroups of $G$, 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. For instance, a group is locally cyclic if and only if its lattice of subgroups is distributive.

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.

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 $mathbf\{Z\}/2mathbf\{Z\}\; *\; mathbf\{Z\}/2mathbf\{Z\}$ is generated by two torsion elements, but is infinite and contains elements of infinite order.

See also

• Zassenhaus lemma, an isomorphism between certain quotients in the lattice of subgroups
• Complemented group, a group with a complemented lattice of subgroups
• Lattice theorem, a Galois connection between the lattice of subgroups of a group and of its quotient

References

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• Review by Ralph Freese in Bull. AMS 33 (4): 487–492.

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External links

• PlanetMath entry on lattice of subgroups

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