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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 large subgroup. 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.## Examples

### Virtually abelian

The following groups are virtually abelian.### Virtually nilpotent

### Virtually polycyclic

### Virtually free

### Others

The free group F_{n} on n generators is virtually F_{2} for any n ≥ 2.
## References

Common uses for this would be when P is abelian, nilpotent, or free.

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.

- Any abelian group.
- The semidirect product $Grtimes\; A$ where G is finite and A is abelian.
- A finite group G (since the trivial subgroup is abelian).

- Any group that is virtually abelian.
- Any nilpotent group.
- The semidirect product $Grtimes\; A$ where G is finite and A is abelian.

- Any free group.
- The semidirect product $Grtimes\; A$ where G is finite and A is free.

- Muller, T. "Combinatorial Aspects of Finitely Generated Virtually Free Groups".
*Journal of the London Mathematical Society*s2-44 (1): 75–94.

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Last updated on Saturday September 06, 2008 at 08:36:25 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Saturday September 06, 2008 at 08:36:25 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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