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382:. Although polycyclic-by-finite groups need not be solvable, they still have many of the finiteness properties of polycyclic groups; for example, they satisfy the maximal condition, and they are finitely presented and
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Examples of polycyclic groups include finitely generated abelian groups, finitely generated
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These groups are particularly interesting because they are the only known examples of
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polycyclic subgroup of finite index, and therefore such groups are also called
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with cyclic factors, that is a finite set of subgroups, let's say
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of a polycyclic group is also such a group of integer matrices.
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is a polycyclic-by-finite group, then the Hirsch length of
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is the number of infinite factors in its subnormal series.
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Ivanov, S. V. (1989), "Group rings of
Noetherian groups",
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is a group that has a polycyclic subgroup of finite
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431:), or group rings of finite injective dimension.
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575:Dmitriĭ Alekseevich Suprunenko, K. A. Hirsch,
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307:proved that solvable subgroups of the integer
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514:Akademiya Nauk SSSR. Matematicheskie Zametki
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395:harv error: no target: CITEREFScott1964 (
374:property. Such a group necessarily has a
168:is polycyclic if and only if it admits a
117:Learn how and when to remove this message
144:that satisfies the maximal condition on
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462:is the Hirsch length of a polycyclic
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291:of a cyclic group by a cyclic group.
129:Type of solvable group in mathematics
303:groups, and finite solvable groups.
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148:(that is, every subgroup is
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599:Encyclopedia of Mathematics
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380:polycyclic-by-finite groups
358:Polycyclic-by-finite groups
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283:is a polycyclic group with
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364:virtually polycyclic group
323:Strongly polycyclic groups
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164:Equivalently, a group
624:Properties of groups
540:Scott, W.R. (1987),
401:and some papers, an
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579:(1976), pp. 174–5;
500:Supersolvable group
333:strongly polycyclic
327:A polycyclic group
594:"Polycyclic group"
548:, pp. 45–46,
546:Dover Publications
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150:finitely generated
66:"Polycyclic group"
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581:Google Books
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542:Group Theory
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44:verification
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477:has finite
429:Ivanov 1989
425:group rings
267:(for every
233:(for every
160:Terminology
134:mathematics
618:Categories
506:References
422:Noetherian
391:Scott 1964
188:such that
77:newspapers
604:EMS Press
526:0025-567X
393:, Ch 7.1)
317:holomorph
301:nilpotent
289:extension
146:subgroups
107:June 2008
489:See also
473:, where
295:Examples
606:, 2001
534:1051052
403:M-group
372:virtual
179:, ...,
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563:Notes
479:index
368:index
263:is a
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439:The
397:help
275:- 1)
241:- 1)
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454:If
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132:In
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530:MR
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