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 groups, the adverb virtually is used to modify a property so that it need only hold for 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
Abelian may refer to:
Mathematics Group theory
* Abelian group, a group in which the binary operation is commutative
** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms
* Metabelian group, a grou ...
,
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
Free may refer to:
Concept
* Freedom, having the ability to do something, without having to obey anyone/anything
* Freethought, a position that beliefs should be formed only on the basis of logic, reason, and empiricism
* Emancipate, to procur ...
. 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 am ...
, 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.
*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 , 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.
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 , volume=159 , pages=159–167 , year=1978 , doi=10.1007/bf01214488
Group theory