For a definition of the term "virtually", see the Wiktionary entry virtually.
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 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 virtuallyH 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.
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...
Virtually Famous is a British comedy panel game show most recently presented by Chris Ramsey. In each episode, two teams of three panelists compete. The...
Virtually Normal: An Argument About Homosexuality (1995; second edition 1996) is a book about the politics of homosexuality by the political commentator...
be virtually Haken. The Virtually Haken conjecture asserts that every compact, irreducible 3-manifold with infinite fundamental group is virtually Haken...
Virtually Heroes is a 2013 American low-budget gamer film directed by G.J. Echternkamp and starring Robert Baker and Mark Hamill. Produced by Roger Corman...
Virtually a Virgin (Hungarian: Majdnem szüz) is a 2008 Hungarian drama film written and directed by Péter Bacsó. It was entered into the 30th Moscow International...
A virtually safe dose (VSD) may be determined for those carcinogens not assumed to have a threshold. Virtually safe doses are calculated by regulatory...
A virtually imaged phased array (VIPA) is an angular dispersive device that, like a prism or a diffraction grating, splits light into its spectral components...
certain finite set. Every cyclic group is virtually cyclic, as is every finite group. An infinite group is virtually cyclic if and only if it is finitely generated...
people or "CAVE dwellers" serves as an acronym for "citizens against virtually everything." It is a pejorative term for citizens who regularly oppose...
Virtually Live is a patented media method and system that uses visualization technology and telemetry to enable a fan experience of virtually attending...
Any subgroup of a strongly polycyclic group is strongly polycyclic. A virtually polycyclic group is a group that has a polycyclic subgroup of finite index...
virtually Haken conjecture states that every compact, orientable, irreducible three-dimensional manifold with infinite fundamental group is virtually...
overseas department or overseas region, although the second is still virtually unused by French media. The following have overseas region status: in...
Office, Jay Ashcroft, Secretary of State. The Missouri Digital Heritage Initiative replaces the Virtually Missouri project. Missouri Digital Heritage...
the Sierra Leone Creole people) is the most widely spoken language in virtually all parts of Sierra Leone. As the Krio language is spoken by 96% of the...
is not socially acceptable, or is considered undesirable,[why?] and is virtually non-existent. Status: Contentious Homeschooling is currently permitted...
instead of through a connecting rod system or similar means. Steam turbines virtually replaced reciprocating engines in electricity generating stations early...
In the mathematical subfield of 3-manifolds, the virtually fibered conjecture, formulated by American mathematician William Thurston, states that every...
Resource Locator (URL), such as https://en.wikipedia.org/, into the browser. Virtually all URLs on the Web start with either http: or https: which means they...
computer using a variety of media. Hard disk drives (HDDs) are found in virtually all older computers, due to their high capacity and low cost, but solid-state...
adequate to describe the process. Image-making seems to have been common to virtually all human cultures since at least the Paleolithic era. Prehistoric examples...
Korean hanja, still used to a certain extent in South Korea, remain virtually identical to traditional characters, with variations between the two forms...
clock faces.[citation needed] Other digits (such as Eastern Arabic) were virtually unknown.[citation needed] Prior to the introduction of Arabic numerals...
represents a voiced dental nasal /n̪/ or voiced alveolar nasal /n/ in virtually all languages that use the Latin alphabet. In many languages, these nasal...
Tramp character). Animated cartoons came principally as short subjects. Virtually all major film production companies had units assigned to develop and...
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