Definitions

Virtually

Virtually

[vur-choo-uh-lee]
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 HG 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

Others

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

References

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