In mathematics, especially in the area of abstract algebra 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. Given a property P, the group G is said to be virtually P if there is a finite index subgroup H \le 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. 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.

Virtually 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

It follows from Stalling's theorem that any torsion-free virtually free group is free.

Others

The free group F_2 on 2 generators is virtually F_n for any n\ge 2 as a consequence of the Nielsen–Schreier theorem and the Schreier index formula. The group is virtually connected as has index 2 in it.