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 subgroup of finite index. 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, solvable or free. For example, virtually solvable groups are one of the two alternatives in the Tits alternative, 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 to H.
A consequence of this is that a finite group is virtually trivial.
The following groups are virtually abelian.
It follows from Stalling's theorem that any torsion-free virtually free group is free.
The free group F2 on 2 generators is virtually Fn for any n ≥ 2 as a consequence of the Nielsen–Schreier theorem and the Schreier index formula.
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