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In particular, a group is virtually trivial if and only if it is finite. Two groups are virtually equal if and only if they are
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says that a finitely generated group is virtually nilpotent if and only if it has polynomial growth.
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107:. For example, virtually solvable groups are one of the two alternatives in the
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732:
Schneebeli, Hans Rudolf (1978). "On virtual properties and group extensions".
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20:
122:
This terminology is also used when P is just another group. That is, if
43:
747:
19:
For a definition of the term "virtually", see the
Wiktionary entry
119:
are precisely the finitely generated virtually nilpotent groups.
42:
is used to modify a property so that it need only hold for a
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Any finite group (since the trivial subgroup is abelian).
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that any torsion-free virtually free group is free.
544:{\displaystyle \operatorname {PSL} (2,\mathbb {Z} )}
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115:states that the finitely generated groups with
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185:The following groups are virtually abelian.
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91:Common uses for this would be when P is
715:{\displaystyle \operatorname {SO} (n)}
683:{\displaystyle \operatorname {O} (n)}
7:
283:Any group that is virtually abelian.
58:if there is a finite index subgroup
508:are both finite. (For example, the
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14:
50:. Given a property P, the group
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594:on 2 generators is virtually
229:is finite. (For example, any
30:, especially in the area of
394:Any virtually cyclic group.
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690:is virtually connected as
453:{\displaystyle N\rtimes H}
416:{\displaystyle N\rtimes H}
377:virtually polycyclic group
374:
345:{\displaystyle N\rtimes H}
308:{\displaystyle N\rtimes H}
255:{\displaystyle N\rtimes H}
231:generalized dihedral group
214:{\displaystyle N\rtimes H}
18:
735:Mathematische Zeitschrift
649:Nielsen–Schreier theorem
647:as a consequence of the
640:{\displaystyle n\geq 2}
434:Any semidirect product
397:Any semidirect product
326:Any semidirect product
289:Any semidirect product
236:Any semidirect product
77:{\displaystyle H\leq G}
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653:Schreier index formula
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614:{\displaystyle F_{n}}
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587:{\displaystyle F_{2}}
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371:Virtually polycyclic
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286:Any nilpotent group.
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16:Mathematical concept
722:has index 2 in it.
493:{\displaystyle H*K}
278:Virtually nilpotent
149:of finite index in
748:10.1007/bf01214488
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557:Stalling's theorem
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194:semidirect product
189:Any abelian group.
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319:is nilpotent and
181:Virtually abelian
117:polynomial growth
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555:It follows from
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365:Gromov's theorem
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130:are groups then
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109:Tits alternative
88:has property P.
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32:abstract algebra
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567:The free group
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145:has a subgroup
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36:infinite groups
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464:is finite and
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382:Virtually free
380:
375:Main article:
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356:is finite and
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266:is finite and
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54:is said to be
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510:modular group
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360:is nilpotent.
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170:commensurable
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38:, the adverb
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34:that studies
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22:
773:Group theory
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473:free product
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427:is free and
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90:
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25:
742:: 159–167.
270:is abelian.
56:virtually P
28:mathematics
756:0358.20048
726:References
658:The group
431:is finite.
389:free group
323:is finite.
159:isomorphic
153:such that
84:such that
46:of finite
701:
669:
632:≥
522:
485:∗
445:⋊
408:⋊
337:⋊
300:⋊
247:⋊
206:⋊
136:virtually
97:nilpotent
69:≤
40:virtually
21:virtually
767:Category
651:and the
621:for any
500:, where
468:is free.
176:Examples
111:, while
101:solvable
44:subgroup
93:abelian
754:
563:Others
460:where
423:where
352:where
315:where
262:where
221:where
48:index
504:and
471:Any
387:Any
192:Any
126:and
105:free
752:Zbl
744:doi
740:159
519:PSL
161:to
157:is
141:if
134:is
103:or
26:In
769::
750:.
738:.
698:SO
655:.
551:.)
233:.)
172:.
165:.
99:,
95:,
758:.
746::
710:)
707:n
704:(
678:)
675:n
672:(
666:O
635:2
629:n
607:n
603:F
580:2
576:F
539:)
535:Z
531:,
528:2
525:(
506:K
502:H
488:K
482:H
466:H
462:N
448:H
442:N
429:H
425:N
411:H
405:N
391:.
358:H
354:N
340:H
334:N
321:H
317:N
303:H
297:N
268:H
264:N
250:H
244:N
227:H
223:N
209:H
203:N
163:H
155:K
151:G
147:K
143:G
139:H
132:G
128:H
124:G
86:H
72:G
66:H
52:G
23:.
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