Virtually

Virtually

In mathematics, especially in the area of abstract algebra which studies infinite groups, the adverb virtually is used to modify a property so that it need only hold for a large subgroup. Given a property P, the group "G" is said to be "virtually P" if there is a finite index subgroup "H"≤"G" such that "H" has property P.

Common uses for this would be when P is abelian, nilpotent, or free.

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 to "H".

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

Examples

Virtually abelian

The following groups are virtually abelian.
*Any abelian group.
*The semidirect product G times A where "G" is finite and "A" is abelian.
*A finite group "G" (since the trivial subgroup is abelian).

Virtually nilpotent

*Any group that is virtually abelian.
*Any nilpotent group.
*The semidirect product G times A where "G" is finite and "A" is abelian.

Virtually polycyclic

Virtually free

*Any free group.
*The semidirect product G times A where "G" is finite and "A" is free.

Others

The free group "F""n" on "n" generators is virtually "F"2 for any "n" ≥ 2.

References

*cite journal|last = Muller|first = T.|title = Combinatorial Aspects of Finitely Generated Virtually Free Groups|journal = Journal of the London Mathematical Society|volume = s2-44|issue = 1|date = 1991|pages = 75–94|url = http://jlms.oxfordjournals.org/cgi/content/long/s2-44/1/75|doi = 10.1112/jlms/s2-44.1.75


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  • virtually — [vʉr′cho͞o ə lē] adv. in effect, although not in fact; for all practical purposes [virtually identical] …   English World dictionary

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  • virtually — [adv] for all practical purposes around, as good as, basically, effectually, essentially, for all intents and purposes*, fundamentally, give or take a little*, guesstimate*, implicitly, in all but name*, in effect, in essence, in substance, in… …   New thesaurus

  • virtually — ► ADVERB 1) nearly; almost. 2) Computing by means of virtual reality techniques …   English terms dictionary

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  • virtually */*/*/ — UK [ˈvɜː(r)tʃʊəlɪ] / US [ˈvɜrtʃuəlɪ] adverb used for emphasizing that a statement is almost completely true Virtually all the students live in university halls of residence. Vaccines have virtually eliminated many childhood diseases. virtually… …   English dictionary

  • virtually — [[t]vɜ͟ː(r)tʃuəli[/t]] ♦♦♦ ADV: ADV with group You can use virtually to indicate that something is so nearly true that for most purposes it can be regarded as true. Virtually all cooking was done over coal fired ranges... It would have been… …   English dictionary

  • virtually — adverb 1. in essence or effect but not in fact (Freq. 4) the strike virtually paralyzed the city I m virtually broke • Derived from adjective: ↑virtual 2. (of actions or states) slightly short of or not quite accomplished; all but (Freq. 4) …   Useful english dictionary

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