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Reverted good faith edit(s) by 79.95.167.79 using STiki

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[[File:Dih4 subgroups (cycle graphs).svg|thumb|400px|[[Hasse diagram]] of the '''l.o.s.''' of the [[dihedral group]] [[Dihedral group of order 8|Dih4]], with the subgroups represented by their [[Cycle graph (algebra)|cycle graphs]]]]
In [[mathematics]], the '''lattice of subgroups''' of a [[Group (mathematics)|group]] <math>G</math> is the [[Lattice (order)|lattice]] whose elements are the [[subgroup]]s of <math>G</math>, with the [[partial order]] [[Relation (mathematics)|relation]] being [[set inclusion]].
In this lattice, the join of two subgroups is the subgroup [[generating set of a group|generated]] by their [[union (set theory)|union]], and the meet of two subgroups is their [[intersection (set theory)|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 {{harvs|first=Øystein|last=Ore|authorlink=Øystein Ore|year=1937|year2=1938|txt}}. For instance, as Ore proved, a group is [[Locally cyclic group|locally cyclic]] if and only if its lattice of subgroups is [[Distributive lattice|distributive]]. Lattice-theoretic characterizations of this type also exist for [[solvable group]]s and [[perfect group]]s {{harv|Suzuki|1951}}.

== Example ==

The [[dihedral group]] [[Dihedral group of order 8|Dih4]] has ten subgroups, counting itself and the [[Trivial group|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 [[Klein four-group|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" ''N''5 as a sublattice.

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

* [[nilpotent group|Nilpotent]] [[normal subgroup]]s form a lattice, which is (part of) the content of [[Fitting's theorem]].
* In general, for any Fitting class ''F'', both the [[subnormal subgroup|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 group]]s. A class of groups is called a Fitting class if it is closed under isomorphism, subnormal subgroups, and products of subnormal subgroups.
* [[Center (group)|Central]] subgroups form a lattice.

However, neither finite subgroups nor torsion subgroups form a lattice: for instance, the [[free product]] <math>\mathbf{Z}/2\mathbf{Z} * \mathbf{Z}/2\mathbf{Z}</math> 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
* Example: [[v:Symmetric_group_S4#Lattice_of_subgroups|Lattice of subgroups of the symmetric group S4]]

== References ==

*{{cite journal
| title = The significance of the system of subgroups for the structure of the group
| author = [[Reinhold Baer|Baer, Reinhold]]; [[Felix Hausdorff|Hausdorff, Felix]]
| journal = [[American Journal of Mathematics]]
| volume = 61
| issue = 1
| year = 1939
| pages = 1–44
| doi = 10.2307/2371383
| jstor = 2371383
| publisher = The Johns Hopkins University Press
| ref = harv}}

*{{Cite journal
| last = Ore | first = Øystein | author-link = Øystein Ore
| doi = 10.1215/S0012-7094-37-00311-9
| mr = 1545977
| issue = 2
| journal = Duke Mathematical Journal
| pages = 149–174
| title = Structures and group theory. I
| volume = 3
| year = 1937
| ref = harv
| postscript = }}.

*{{Cite journal
| last = Ore | first = Øystein | author-link = Øystein Ore
| doi = 10.1215/S0012-7094-38-00419-3
| mr = 1546048
| issue = 2
| journal = Duke Mathematical Journal
| pages = 247–269
| title = Structures and group theory. II
| volume = 4
| year = 1938
| ref = harv
| postscript = }}.

*{{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
| ref = harv}}

*{{Cite book
| last = Schmidt | first = Roland
| title = Subgroup Lattices of Groups
| year = 1994
| series = Expositions in Math
| volume = 14
| publisher = Walter de Gruyter
| isbn = 978-3-11-011213-9
| ref = harv
| postscript = }}. [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
| jstor = 1990375
| publisher = American Mathematical Society
| ref = harv}}

*{{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
| pages = 400–412
| ref = harv}}

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

[[Category:Lattice theory]]
[[Category:Group theory]]

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en>Jesse V.: Reverted good faith edit(s) by 79.95.167.79 using STiki