Lattice of subgroups

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

*cite journal
title = The significance of the system of subgroups for the structure of the group
author = Baer, Reinhold; Hausdorff, Felix
journal = American Journal of Mathematics
volume = 61
issue = 1
year = 1939
pages = 1–44
doi = 10.2307/2371383

*cite journal
author = Rottlaender, Ada
title = Nachweis der Existenz nicht-isomorpher Gruppen von gleicher Situation der Untergruppen
journal = Mathematische Zeitschrift
volume = 28
issue = 1
year = 1928
pages = 641–653
doi = 10.1007/BF01181188

*cite book
author = Schmidt, Roland
title = Subgroup Lattices of Groups
year = 1994
publisher = Expositions in Math, vol. 14, de Gruyter [http://www.ams.org/bull/1996-33-04/S0273-0979-96-00676-3/S0273-0979-96-00676-3.pdf Review] by Ralph Freese in Bull. AMS 33 (4): 487–492.

*cite journal
title = On the lattice of subgroups of finite groups
author = Suzuki, Michio
authorlink = Michio Suzuki
journal = Transactions of the American Mathematical Society
volume = 70
issue = 2
year = 1951
pages = 345–371
doi = 10.2307/1990375

*cite book
author = Suzuki, Michio
authorlink = Michio Suzuki
title = Structure of a Group and the Structure of its Lattice of Subgroups
publisher = Springer Verlag
location = Berlin
year = 1956

*cite journal
author = Yakovlev, B. V.
title = Conditions under which a lattice is isomorphic to a lattice of subgroups of a group
journal = Algebra and Logic
volume = 13
issue = 6
year = 1974
doi = 10.1007/BF01462952

External links

* [http://planetmath.org/encyclopedia/LatticeOfSubgroups.html PlanetMath entry on lattice of subgroups]