In mathematics Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions, especially in the area of abstract algebra Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. The phrase abstract algebra was coined at the turn of the 20th century to distinguish this area from what was normally referred to as algebra, the study of the rules for manipulating formulae which studies infinite groups, the adverb virtually is used to modify a property so that it need only hold for a subgroup In the mathematical subject known as 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 x H is a group operation on H. This is usually represented notationally by H ≤ G, read as of finite index. Given a property P, the group G is said to be virtually P if there is a finite index In mathematics, specifically group theory, the index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" of H that fill up G. For example, if H has index 2 in G, then intuitively "half" of the elements of G lie in H. The index of H in G is usually denoted |G : H| or [G : subgroup H≤G such that H has property P.

Common uses for this would be when P is abelian An abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers. They are named after Niels Henrik Abel, nilpotent, or free In mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many elements of S and their inverses.

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 abstract algebra, an isomorphism is a bijective map f such that both f and its inverse f −1 are homomorphisms, i.e., structure-preserving mappings. In the more general setting of category theory, an isomorphism is a morphism f: X → Y in a category for which there exists an "inverse" f −1: Y → X, with the property that both f ∠to H.

A consequence of this is that a finite group is virtually trivial.

Contents

Examples

Virtually abelian

The following groups are virtually abelian.

Virtually nilpotent

Virtually polycyclic

Main article: virtually polycyclic group

Virtually free

Others

The free group Fn on n generators is virtually F2 for any n ≥ 2.

References

Categories: Group theory In mathematics, a group is a set, together with a binary operation satisfying certain axioms, detailed in the group article

Personal tools
Namespaces
">
Variants
Views
">
Actions
Search">
The Central London Railway was a railway company established in 1889 to construct a deep-level underground "tube" railway in London. Funding for construction was obtained in 1895 through a syndicate of financiers and construction work took place from 1896 to 1900. When opened in 1900, the railway served 13 stations and ran completely
Navigation
Interaction
Toolbox
Print/export

 

The above information uses material from Wikipedia and is licensed under the GNU Free Documentation License The purpose of this License is to make a manual, textbook, or other functional and useful document "free" in the sense of freedom: to assure everyone the effective freedom to copy and redistribute it, with or without modifying it, either commercially or noncommercially. Secondarily, this License preserves for the author and publisher a.
Some facts may not have been fully verified for accuracy. [Disclaimers Wikipedia is an online open-content collaborative encyclopedia, that is, a voluntary association of individuals and groups working to develop a common resource of human knowledge. The structure of the project allows anyone with an Internet connection to alter its content. Please be advised that nothing found here has necessarily been reviewed by]
This page was last archived by our server on Sat Jul 31 01:25:35 2010. [ refresh local cache ]
Displaying this page or its contents does not use any Wikimedia Foundation's resources.
The owners of this site proudly support the Wikimedia Foundation.

[Hide]▼
[Hide]▲