Virtually

From Wikipedia, the free encyclopedia

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 it has a subgroup K such that K is isomorphic to H.

A consequence of this is that finite group theory is virtually trivial.

Contents 1 Examples <33 1.1 Virtually Abelian 1.2 Virtually nilpotent 1.3 Virtually free 1.4 Others 2 References

[edit] Examples <33

[edit] Virtually Abelian

The following groups are virtually abelian.

• Any abelian group
• The semidirect product where G is finite and A is abelian.
• A finite group G (since the trivial subgroup is abelian).

[edit] Virtually nilpotent

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

[edit] Virtually free

• Any free group
• The semidirect product where G is finite and A is free.

[edit] Others

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

[edit] References

• Muller, T. (1991). "Combinatorial Aspects of Finitely Generated Virtually Free Groups". Journal of the London Mathematical Society s2-44 (1): 75–94. doi:10.1112/jlms/s2-44.1.75.