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.
Famous quotes containing the word virtually:
“Any historian of the literature of the modern age will take virtually for granted the adversary intention, the actually subversive intention, that characterizes modern writing—he will perceive its clear purpose of detaching the reader from the habits of thought and feeling that the larger culture imposes, of giving him a ground and a vantage point from which to judge and condemn, and perhaps revise, the culture that produces him.”
—Lionel Trilling (1905–1975)
“If twins are believed to be less intelligent as a class than single-born children, it is not surprising that many times they are also seen as ripe for social and academic problems in school. No one knows the extent to which these kind of attitudes affect the behavior of multiples in school, and virtually nothing is known from a research point of view about social behavior of twins over the age of six or seven, because this hasn’t been studied either.”
—Pamela Patrick Novotny (20th century)
“The law is equal before all of us; but we are not all equal before the law. Virtually there is one law for the rich and another for the poor, one law for the cunning and another for the simple, one law for the forceful and another for the feeble, one law for the ignorant and another for the learned, one law for the brave and another for the timid, and within family limits one law for the parent and no law at all for the child.”
—George Bernard Shaw (1856–1950)