# Lattice of subgroups

In

mathematics , the**lattice of subgroups**of a group $G$ is the lattice whose elements are thesubgroup s of $G$, with thepartial order relation beingset 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 Dih_{4}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 samecyclic group C_{4}. In addition, there are two groups of the form C_{2}×C_{2}, 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 subgroup s form a lattice, which is (part of) the content ofFitting'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 ofsolvable group s. 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 acomplemented lattice of subgroups

*Lattice theorem , aGalois 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*]

*Wikimedia Foundation.
2010.*

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