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 ${\displaystyle H\leq G}$ 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.