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371:. 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
402:, which by Hirsch's theorem can also be expressed as a group which has a finite length subnormal series with each factor a finite group or an infinite
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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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is infinite. Any subgroup of a strongly polycyclic group is strongly polycyclic.
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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
46:. Unsourced material may be challenged and removed.
420:), or group rings of finite injective dimension.
564:Dmitriĭ Alekseevich Suprunenko, K. A. Hirsch,
296:proved that solvable subgroups of the integer
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503:Akademiya Nauk SSSR. Matematicheskie Zametki
394:refers to what is now called a polycyclic-
384:harv error: no target: CITEREFScott1964 (
363:property. Such a group necessarily has a
157:is polycyclic if and only if it admits a
106:Learn how and when to remove this message
133:that satisfies the maximal condition on
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451:is the Hirsch length of a polycyclic
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280:of a cyclic group by a cyclic group.
118:Type of solvable group in mathematics
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292:groups, and finite solvable groups.
44:adding citations to reliable sources
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31:needs additional citations for
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137:(that is, every subgroup is
588:Encyclopedia of Mathematics
369:polycyclic-by-finite groups
347:Polycyclic-by-finite groups
272:is a polycyclic group with
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353:virtually polycyclic group
312:Strongly polycyclic groups
300:are polycyclic; and later
276:≤ 2, or in other words an
141:). Polycyclic groups are
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201:is the trivial subgroup
436:of a polycyclic group
153:Equivalently, a group
613:Properties of groups
529:Scott, W.R. (1987),
390:and some papers, an
298:general linear group
40:improve this article
568:(1976), pp. 174–5;
489:Supersolvable group
322:strongly polycyclic
316:A polycyclic group
583:"Polycyclic group"
537:, pp. 45–46,
535:Dover Publications
359:, an example of a
143:finitely presented
139:finitely generated
55:"Polycyclic group"
544:978-0-486-65377-8
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29:This article
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570:Google Books
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531:Group Theory
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509:(6): 61–66,
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484:Group theory
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38:Please help
33:verification
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466:has finite
418:Ivanov 1989
414:group rings
256:(for every
222:(for every
149:Terminology
123:mathematics
607:Categories
495:References
411:Noetherian
380:Scott 1964
177:such that
66:newspapers
593:EMS Press
515:0025-567X
382:, Ch 7.1)
306:holomorph
290:nilpotent
278:extension
135:subgroups
96:June 2008
478:See also
462:, where
284:Examples
595:, 2001
523:1051052
392:M-group
361:virtual
168:, ...,
80:scholar
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365:normal
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552:Notes
468:index
357:index
252:is a
129:is a
87:JSTOR
73:books
539:ISBN
511:ISSN
428:The
386:help
264:- 1)
230:- 1)
125:, a
59:news
470:in
458:of
443:If
432:or
121:In
42:by
609::
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519:MR
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507:46
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396:by
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351:A
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332:+1
268:A
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241:+1
220:+1
572:.
472:G
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460:G
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438:G
416:(
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340:i
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274:n
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199:0
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186:n
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170:G
166:0
163:G
155:G
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