Virtually

For the definitions of this word, see the Wiktionary definition of 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 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.

Examples

Virtually abelian

The following groups are virtually abelian.

• Any abelian group.
• Any semidirect product $N\rtimes H$ where N is abelian and H is finite. (For example, any generalized dihedral group.)
• Any semidirect product $N\rtimes H$ 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 $N\rtimes H$ where N is nilpotent and H is finite.
• Any semidirect product $N\rtimes H$ where N is finite and H is nilpotent.

Virtually polycyclic

Virtually free

• Any free group.
• Any virtually cyclic group.
• Any semidirect product $N\rtimes H$ where N is free and H is finite.
• Any semidirect product $N\rtimes H$ where N is finite and H is free.
• Any free product H * K, where H and K are both finite. (For example, the modular group PSL(2,Z).)

It follows from Stalling's theorem that any torsion-free virtually free group is free.

Others

The free group F₂ on 2 generators is virtually F_n for any n ≥ 2 as a consequence of the Nielsen–Schreier theorem and the Schreier index formula.

References

• Schneebeli, Hans Rudolf (1978). "On virtual properties and group extensions". Math. Z. 159: 159–167. doi:10.1007/bf01214488. Zbl 0358.20048.

External links

• Schreier index formula at PlanetMath.