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