The lattice of subgroups of the dihedral group Dih₄, represented as groups of rotations and reflections of a plane figure. The lattice is shown as a Hasse diagram.

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. This article may be confusing for some readers, and should be edited to enhance clarity. ... In the mathematical area of order theory, a Hasse diagram (pronounced HAHS uh, named after Helmut Hasse (1898–1979)) is a simple picture of a finite partially ordered set. ... Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... The name lattice is suggested by the form of the Hasse diagram depicting it. ... In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group... In mathematics, a partially ordered set (or poset for short) is a set equipped with a special binary relation which formalizes the intuitive concept of an ordering. ... In mathematics, a finitary relation is defined by one of the formal definitions given below. ... In abstract algebra, a generating set of a group is a subset S such that every element of G can be expressed as the product of finitely many elements of S and their inverses. ... In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ... In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. ...

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. In mathematics, a locally cyclic group is a group (G, *) in which every finitely generated subgroup is cyclic. ... In mathematics, distributive lattices are lattices for which the operations of join and meet distribute over each other. ...

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 article may be confusing for some readers, and should be edited to enhance clarity. ... The following list in mathematics contains the finite groups of small order up to group isomorphism. ... In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a (or na... This article is about the mathematical group. ...

See also

• Zassenhaus lemma

In mathematics, the butterfly lemma or Zassenhaus lemma is a technical result on the lattice of subgroups of a group. ...

References

• Baer, Reinhold; Hausdorff, Felix (1939). "The significance of the system of subgroups for the structure of the group". American Journal of Mathematics 61 (1): 1–44. DOI:10.2307/2371383.

• Rottlaender, Ada (1928). "Nachweis der Existenz nicht-isomorpher Gruppen von gleicher Situation der Untergruppen". Mathematische Zeitschrift 28 (1): 641–653. DOI:10.1007/BF01181188.

• Schmidt, Roland (1994). Subgroup Lattices of Groups. Expositions in Math, vol. 14, de Gruyter. Review by Ralph Freese in Bull. AMS 33 (4): 487–492.

• Suzuki, Michio (1951). "On the lattice of subgroups of finite groups". Transactions of the American Mathematical Society 70 (2): 345–371. DOI:10.2307/1990375.

• Suzuki, Michio (1956). Structure of a Group and the Structure of its Lattice of Subgroups. Berlin: Springer Verlag.

• Yakovlev, B. V. (1974). "Conditions under which a lattice is isomorphic to a lattice of subgroups of a group". Algebra and Logic 13 (6). DOI:10.1007/BF01462952.

Reinhold Baer Reinhold Baer (July 22, 1902 - October 22, 1979) was a mathematician who introduced injective modules in 1940. ... Felix Hausdorff Felix Hausdorff (November 8, 1868 – January 26, 1942) was a German mathematician who is considered to be one of the founders of modern topology and who contributed significantly to set theory and functional analysis. ... American Journal of Mathematics, April 2006 issue. ... A digital object identifier (or DOI) is a permanent identifier (permalink) given to a World Wide Web file or other Internet document so that if its Internet address changes, users will be redirected to its new address. ... A digital object identifier (or DOI) is a permanent identifier (permalink) given to a World Wide Web file or other Internet document so that if its Internet address changes, users will be redirected to its new address. ... Michio Suzuki (Japanese: 鈴木 通夫 Suzuki Michio; (October 2, 1926 - May 31, 1998) was a Japanese mathematician at the University of Illinois at Urbana-Champaign from 1953 to his death in 1999 who studied group theory. ... Transactions of the American Mathematical Society is a monthly mathematics journal published by the American Mathematical Society. ... A digital object identifier (or DOI) is a permanent identifier (permalink) given to a World Wide Web file or other Internet document so that if its Internet address changes, users will be redirected to its new address. ... Michio Suzuki (Japanese: 鈴木 通夫 Suzuki Michio; (October 2, 1926 - May 31, 1998) was a Japanese mathematician at the University of Illinois at Urbana-Champaign from 1953 to his death in 1999 who studied group theory. ... A digital object identifier (or DOI) is a permanent identifier (permalink) given to a World Wide Web file or other Internet document so that if its Internet address changes, users will be redirected to its new address. ...

External link

• PlanetMath entry on lattice of subgroups