# Virtually

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Virtually
> Markdown URL: https://mediated.wiki/source/Virtually.md
> Source: https://en.wikipedia.org/wiki/Virtually
> Source revision: 1250818690
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

Mathematical concept

For a definition of the term "virtually", see the Wiktionary entry [virtually](https://en.wiktionary.org/wiki/virtually).

In [mathematics](/source/Mathematics), especially in the area of [abstract algebra](/source/Abstract_algebra) that studies [infinite groups](/source/Infinite_group), the adverb **virtually** is used to modify a property so that it need only hold for a [subgroup](/source/Subgroup) of finite [index](/source/Index_of_a_subgroup). Given a property P, the group *G* is said to be *virtually P* if there is a finite index subgroup H ≤ G {\displaystyle H\leq G} such that *H* has property P.

Common uses for this would be when P is [abelian](/source/Abelian_group), [nilpotent](/source/Nilpotent_group), [solvable](/source/Solvable_group) or [free](/source/Free_group). For example, virtually solvable groups are one of the two alternatives in the [Tits alternative](/source/Tits_alternative), while [Gromov's theorem](/source/Gromov's_theorem_on_groups_of_polynomial_growth) states that the finitely generated groups with [polynomial growth](/source/Growth_rate_(group_theory)) 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](/source/Isomorphism) 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](/source/Commensurability_(group_theory)).

## Examples

### Virtually abelian

The following groups are virtually abelian.

- Any abelian group.

- Any [semidirect product](/source/Semidirect_product) N ⋊ H {\displaystyle N\rtimes H} where *N* is abelian and *H* is finite. (For example, any [generalized dihedral group](/source/Generalized_dihedral_group).)

- Any semidirect product N ⋊ H {\displaystyle N\rtimes H} where *N* is finite and *H* is abelian.

- Any finite group (since the trivial subgroup is abelian).

### Virtually nilpotent

- Any group that is virtually abelian.

- Any nilpotent group.

- Any semidirect product N ⋊ H {\displaystyle N\rtimes H} where *N* is nilpotent and *H* is finite.

- Any semidirect product N ⋊ H {\displaystyle N\rtimes H} where *N* is finite and *H* is nilpotent.

[Gromov's theorem](/source/Gromov's_theorem_on_groups_of_polynomial_growth) says that a finitely generated group is virtually nilpotent if and only if it has polynomial growth.

### Virtually polycyclic

Main article: [virtually polycyclic group](/source/Virtually_polycyclic_group)

### Virtually free

- Any [free group](/source/Free_group).

- Any finite group (since the trivial subgroup is the free group on the empty set of generators).

- Any virtually [cyclic group](/source/Cyclic_group). (Either it is finite in which case it falls into the above case, or it is infinite and contains Z {\displaystyle \mathbb {Z} } as a subgroup.)

- Any semidirect product N ⋊ H {\displaystyle N\rtimes H} where *N* is free and *H* is finite.

- Any semidirect product N ⋊ H {\displaystyle N\rtimes H} where *N* is finite and *H* is free.

- Any [free product](/source/Free_product) H ∗ K {\displaystyle H*K} , where *H* and *K* are both finite. (For example, the [modular group](/source/Modular_group) PSL ⁡ ( 2 , Z ) {\displaystyle \operatorname {PSL} (2,\mathbb {Z} )} .)

It follows from [Stalling's theorem](/source/Stallings_theorem_about_ends_of_groups#Applications_and_generalizations) that any torsion-free virtually free group is free.

### Others

The free group F 2 {\displaystyle F_{2}} on 2 generators is virtually F n {\displaystyle F_{n}} for any n ≥ 2 {\displaystyle n\geq 2} as a consequence of the [Nielsen–Schreier theorem](/source/Nielsen%E2%80%93Schreier_theorem) and the [Schreier index formula](/source/Schreier_index_formula).

The group O ⁡ ( n ) {\displaystyle \operatorname {O} (n)} is virtually connected as SO ⁡ ( n ) {\displaystyle \operatorname {SO} (n)} has index 2 in it.

## References

Look up ***[virtually](https://en.wiktionary.org/wiki/virtually)*** in Wiktionary, the free dictionary.

- Schneebeli, Hans Rudolf (1978). "On virtual properties and group extensions". *[Mathematische Zeitschrift](/source/Mathematische_Zeitschrift)*. **159**: 159–167. [doi](/source/Doi_(identifier)):[10.1007/bf01214488](https://doi.org/10.1007%2Fbf01214488). [Zbl](/source/Zbl_(identifier)) [0358.20048](https://zbmath.org/?format=complete&q=an:0358.20048).

---
Adapted from the Wikipedia article [Virtually](https://en.wikipedia.org/wiki/Virtually) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Virtually?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.