AskDefine | Define virtually

Dictionary Definition

virtually adv
1 (intensifier before a figurative expression) without exaggeration; "our eyes were literally pinned to TV during the Gulf war" [syn: literally]
2 in essence or effect but not in fact; "the strike virtually paralyzed the city"; "I'm virtually broke"
3 (of actions or states) slightly short of or not quite accomplished; `near' is sometimes used informally for `nearly' and `most' is sometimes used informally for `almost'; "the job is (just) about done"; "the baby was almost asleep when the alarm sounded"; "we're almost finished"; "the car all but ran her down"; "he nearly fainted"; "talked for nigh onto 2 hours"; "the recording is well-nigh perfect"; "virtually all the parties signed the contract"; "I was near exhausted by the run"; "most everyone agrees" [syn: about, just about, almost, most, all but, nearly, near, nigh, well-nigh]

User Contributed Dictionary

English

Pronunciation

Adverb

virtually
  1. practically; almost but not quite
  2. Without exaggeration; literally
  3. In essence, but not in fact

Translations

practically
literally
in essence, but not in fact

Extensive Definition

Given a property P, the group G is said to be virtually P if there is a finite index subgroup H 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.

Examples <33

Virtually Abelian

The following groups are virtually abelian.
  • Any abelian group
  • The semidirect product G\rtimes 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 G\rtimes A where G is finite and A is abelian.

Virtually free

Others

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

References

  • ">http://jlms.oxfordjournals.org/cgi/content/long/s2-44/1/75}}

Synonyms, Antonyms and Related Words

absolutely, actually, all in all, almost, almost entirely, approximately, as good as, basically, by and large, chiefly, effectively, effectually, essentially, fundamentally, in effect, in essence, in substance, in the main, latently, mainly, morally, more or less, mostly, nearly, on balance, on the whole, potentially, practically, substantially, underlyingly
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