adv.

実質的に

関: materially、substantial、substantially、virtual

WordNet

in essence or effect but not in fact; "the strike virtually paralyzed the city"; "Im virtually broke"
having a firm basis in reality and being therefore important, meaningful, or considerable; "substantial equivalents" (同)substantive
having substance or capable of being treated as fact; not imaginary; "the substantial world"; "a mere dream, neither substantial nor practical"; "most ponderous and substantial things"- Shakespeare (同)real, material
with respect to material aspects; "psychologically similar but materially different"
to a significant degree; "it aided him materially in winning the argument"
in a strong substantial way; "the house was substantially built"
existing in essence or effect though not in actual fact; "a virtual dependence on charity"; "a virtual revolution"; "virtual reality"
being actually such in almost every respect; "a practical failure"; "the once elegant temple lay in virtual ruin" (同)practical

PrepTutorEJDIC

事実上,実質的には
『かなりの』,相当の,十分満足できる / 『がんじょうな』,じょうぶな,曳い / 資産のある / (架空でなく)『実在する』,実体のある / 『実質上の』,実質的な
物質的に,物理的に;肉体的に / 実質的に / 大いに,著しく
概して,本質的に / 十分に,豊富に
(名目上・表面上はそうではないが)事実上の,実際上の

Wikipedia preview

出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2017/08/27 20:25:05」(JST)

wiki en

[Wiki en表示]

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. Zbl 0358.20048. doi:10.1007/bf01214488.

External links

Schreier index formula at PlanetMath.

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★リンクテーブル★

リンク元	「substantial」「materially」「substantially」「実質的」
関連記事	「virtual」

「substantial」

　　[★]

adj.

実内容のある(食事など)。かなりの、たいした。しっかりした、丈夫な、堅固な。重要な、価値のある。資産のある、裕福な。金銭上の信用のある。実力のある

occasionally the blood loss is more substantial or the split in the wall may be deeper than just the mucosa, leading to perforation.

A large pulmonary embolism results in both hypoxemia and a substantial increase in right ventricular afterload.(PHD.240)

実質の、実体の、実在する、本当の

n.

関: appreciable, appreciably, considerable, considerably, correspond, equal, equivalent, fairly, materially, quite, sizeable, substantially, virtual, virtually

「materially」

　　[★]

adv.

実質的に。大いに、著しく(considerably)。

Since most events occurred in a small number of recent trials that used secure randomisation methods and treatment blinding, sensitivity analyses (available on request) indicated that our results were not materially influenced by uncertainties about the quality of older trials.