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 ...
, especially in the area of
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...
that studies
infinite group
In group theory, an area of mathematics, an infinite group is a group whose underlying set contains an infinite number of elements. In other words, it is a group of infinite order.
Examples
* (Z, +), the group of integers with addition is infi ...
s, the adverb virtually is used to modify a property so that it need only hold for a
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 ...
of finite
index
Index (or its plural form indices) may refer to:
Arts, entertainment, and media Fictional entities
* Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index''
* The Index, an item on a Halo megastru ...
. Given a property P, the group ''G'' is said to be ''virtually P'' if there is a finite index subgroup
such that ''H'' has property P.
Common uses for this would be when P is
abelian,
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 class ...
,
solvable or
free. For example, virtually solvable groups are one of the two alternatives in the
Tits alternative In mathematics, the Tits alternative, named for Jacques Tits, is an important theorem about the structure of finitely generated linear groups.
Statement
The theorem, proven by Tits, is stated as follows.
Consequences
A linear group is not a ...
, while
Gromov's theorem states that the finitely generated groups with
polynomial growth are precisely the finitely generated virtually nilpotent groups.
This terminology is also used when P is just another group. That is, if ''G'' and ''H'' are groups then ''G'' is ''virtually'' ''H'' if ''G'' has a subgroup ''K'' of finite index in ''G'' such that ''K'' is
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
to ''H''.
In particular, a group is virtually trivial if and only if it is finite. Two groups are virtually equal if and only if they are
commensurable.
Examples
Virtually abelian
The following groups are virtually abelian.
*Any abelian group.
*Any
semidirect product
In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product:
* an ''inner'' semidirect product is a particular way in w ...
where ''N'' is abelian and ''H'' is finite. (For example, any
generalized dihedral group
In mathematics, the generalized dihedral groups are a family of groups with algebraic structures similar to that of the dihedral groups. They include the finite dihedral groups, the infinite dihedral group, and the orthogonal group ''O''(2). Dih ...
.)
*Any semidirect product
where ''N'' is finite and ''H'' is abelian.
*Any finite group (since the trivial subgroup is abelian).
Virtually nilpotent
*Any group that is virtually abelian.
*Any nilpotent group.
*Any semidirect product
where ''N'' is nilpotent and ''H'' is finite.
*Any semidirect product
where ''N'' is finite and ''H'' is nilpotent.
Gromov's theorem says that a finitely generated group is virtually nilpotent if and only if it has polynomial growth.
Virtually polycyclic
Virtually free
*Any
free group
In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''−1' ...
.
*Any virtually cyclic group.
*Any semidirect product
where ''N'' is free and ''H'' is finite.
*Any semidirect product
where ''N'' is finite and ''H'' is free.
*Any
free product
In mathematics, specifically group theory, the free product is an operation that takes two groups ''G'' and ''H'' and constructs a new The result contains both ''G'' and ''H'' as subgroups, is generated by the elements of these subgroups, and i ...
, where ''H'' and ''K'' are both finite. (For example, the
modular group
In mathematics, the modular group is the projective special linear group of matrices with integer coefficients and determinant 1. The matrices and are identified. The modular group acts on the upper-half of the complex plane by fractional l ...
.)
It follows from
Stalling's theorem that any torsion-free virtually free group is free.
Others
The free group
on 2 generators is virtually
for any
as a consequence of the
Nielsen–Schreier theorem
In group theory, a branch of mathematics, the Nielsen–Schreier theorem states that every subgroup of a free group is itself free. It is named after Jakob Nielsen and Otto Schreier.
Statement of the theorem
A free group may be defined from a grou ...
and the
Schreier index formula Schreier is a surname of German origin. Notable people with the surname include:
*Christian Schreier (born 1959), German footballer
*Dan Moses Schreier, American sound designer and composer
*Jake Schreier, American director
*Józef Schreier, Polis ...
.
The group
is virtually connected as
has index 2 in it.
References
* {{cite journal , last=Schneebeli , first=Hans Rudolf , title=On virtual properties and group extensions , zbl=0358.20048 , journal=
Mathematische Zeitschrift
''Mathematische Zeitschrift'' (German for ''Mathematical Journal'') is a mathematical journal for pure and applied mathematics published by Springer Verlag.
It was founded in 1918 and edited by Leon Lichtenstein together with Konrad Knopp, Erhard ...
, volume=159 , pages=159–167 , year=1978 , doi=10.1007/bf01214488
Group theory