In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the lattice of subgroups of a
group is the
lattice whose elements are the
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
s of
, with the
partial order
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
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
In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, thei ...
.
Example
The
dihedral group
In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, ...
Dih4 has ten subgroups, counting itself and the
trivial subgroup. Five of the eight group elements generate subgroups of order two, and the other two non-identity elements both generate the same
cyclic
Cycle, cycles, or cyclic may refer to:
Anthropology and social sciences
* Cyclic history, a theory of history
* Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr.
* Social cycle, various cycles in so ...
subgroup of order four. In addition, there are two subgroups of the form
Z2 × Z2, 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 the branch of mathematics called order theory, a modular lattice is a lattice that satisfies the following self- dual condition,
;Modular law: implies
where are arbitrary elements in the lattice, ≤ is the partial order, and & ...
in general. Indeed, this particular lattice contains the forbidden "pentagon" ''N''
5 as a
sublattice.
Properties
For any ''A'', ''B'', and ''C'' subgroups of a group with ''A'' ≤ ''C'' (''A'' subgroup of ''C'') then ''AB'' ∩ ''C'' = ''A(B ∩ C)''; the multiplication here is the
product of subgroups In mathematics, one can define a product of group subsets in a natural way. If ''S'' and ''T'' are subsets of a group ''G'', then their product is the subset of ''G'' defined by
:ST = \.
The subsets ''S'' and ''T'' need not be subgroups for this pr ...
. This property has been called the ''modular property of groups'' or ''(
Dedekind's) modular law'' (, ). Since for two normal subgroups the product is actually the smallest subgroup containing the two, the normal subgroups form a
modular lattice
In the branch of mathematics called order theory, a modular lattice is a lattice that satisfies the following self- dual condition,
;Modular law: implies
where are arbitrary elements in the lattice, ≤ is the partial order, and & ...
.
The
Lattice theorem establishes a
Galois connection In mathematics, especially in order theory, a Galois connection is a particular correspondence (typically) between two partially ordered sets (posets). Galois connections find applications in various mathematical theories. They generalize the funda ...
between the lattice of subgroups of a group and that of its quotients.
The
Zassenhaus lemma Zassenhaus is a German surname. Notable people with the surname include:
* Hans Zassenhaus
Hans Julius Zassenhaus (28 May 1912 – 21 November 1991) was a German mathematician, known for work in many parts of abstract algebra, and as a pioneer of ...
gives an isomorphism between certain combinations of quotients and products in the lattice of subgroups.
In general, there is no restriction on the shape of the lattice of subgroups, in the sense that every lattice is isomorphic to a sublattice of the subgroup lattice of some group. Furthermore, every
finite lattice is isomorphic to a sublattice of the subgroup lattice of some
finite group
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or ma ...
.
Characteristic lattices
Subgroups with certain properties form lattices, but other properties do not.
*
Normal subgroup
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G ...
s always form a modular lattice. In fact, the essential property that guarantees that the lattice is modular is that subgroups commute with each other, i.e. that they are
quasinormal subgroup __NOTOC__
In mathematics, in the field of group theory, a quasinormal subgroup, or permutable subgroup, is a subgroup of a group that commutes (permutes) with every other subgroup with respect to the product of subgroups. The term ''quasinormal su ...
s.
*
Nilpotent
In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0.
The term was introduced by Benjamin Peirce in the context of his work on the cl ...
normal subgroups form a lattice, which is (part of) the content of
Fitting's theorem
Fitting's theorem is a mathematical theorem proved by Hans Fitting. It can be stated as follows:
:If ''M'' and ''N'' are nilpotent normal subgroups of a group ''G'', then their product ''MN'' is also a nilpotent normal subgroup of ''G''; if, m ...
.
* 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 group
In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminate ...
s. A class of groups is called a Fitting class if it is closed under isomorphism, subnormal subgroups, and products of subnormal subgroups.
*
Central
Central is an adjective usually referring to being in the center of some place or (mathematical) object.
Central may also refer to:
Directions and generalised locations
* Central Africa, a region in the centre of Africa continent, also known a ...
subgroups form a lattice.
However, neither finite subgroups nor torsion subgroups form a lattice: for instance, the
free product is generated by two torsion elements, but is infinite and contains elements of infinite order.
The fact that normal subgroups form a modular lattice is a particular case of a more general result, namely that in any
Maltsev variety (of which groups are an example), the
lattice of congruences is modular .
Characterizing groups by their subgroup lattices
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 . For instance, as Ore proved, a group is
locally cyclic if and only if its lattice of subgroups is
distributive. If additionally the lattice satisfies the
ascending chain condition In mathematics, the ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some algebraic structures, most importantly ideals in certain commutative rings.Jacobson (2009), p. 142 and 147 These c ...
, then the group is cyclic.
The groups whose lattice of subgroups is a
complemented lattice are called
complemented group In mathematics, in the realm of group theory, the term complemented group is used in two distinct, but similar ways.
In , a complemented group is one in which every subgroup has a group-theoretic complement. Such groups are called completely fact ...
s , and the groups whose lattice of subgroups are
modular lattice
In the branch of mathematics called order theory, a modular lattice is a lattice that satisfies the following self- dual condition,
;Modular law: implies
where are arbitrary elements in the lattice, ≤ is the partial order, and & ...
s are called
Iwasawa group
__NOTOC__
In mathematics, a group is called an Iwasawa group, M-group or modular group if its lattice of subgroups is modular. Alternatively, a group ''G'' is called an Iwasawa group when every subgroup of ''G'' is permutable in ''G'' .
proved ...
s or modular groups . Lattice-theoretic characterizations of this type also exist for
solvable group
In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminate ...
s and
perfect group
In mathematics, more specifically in group theory, a group is said to be perfect if it equals its own commutator subgroup, or equivalently, if the group has no non-trivial abelian quotients (equivalently, its abelianization, which is the unive ...
s .
References
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*
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*
*
*
*
*
*
Reviewby Ralph Freese in Bull. AMS 33 (4): 487–492.
*
*
*
*{{cite journal , last1=Zacher , first1=Giovanni , title=Caratterizzazione dei gruppi risolubili d'ordine finito complementati , url=http://www.numdam.org/item?id=RSMUP_1953__22__113_0 , mr=0057878 , year=1953 , journal=
Rendiconti del Seminario Matematico della Università di Padova , issn=0041-8994 , volume=22 , pages=113–122
External links
PlanetMath entry on lattice of subgroups* Example:
Lattice of subgroups of the symmetric group S4
Lattice theory
Group theory