Virtually

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.

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.

Examples

Virtually abelian

The following groups are virtually abelian.

• 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).

Virtually nilpotent

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

Virtually polycyclic

Virtually free

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

Others

The free group F_n on n generators is virtually F₂ for any n ≥ 2.

References

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