| 词条 | Polycyclic group |
| 释义 |
In mathematics, a polycyclic group is a solvable group that satisfies the maximal condition on subgroups (that is, every subgroup is finitely generated). Polycyclic groups are finitely presented, and this makes them interesting from a computational point of view. TerminologyEquivalently, a group G is polycyclic if and only if it admits a subnormal series with cyclic factors, that is a finite set of subgroups, let's say G0, ..., Gn such that
A metacyclic group is a polycyclic group with n ≤ 2, or in other words an extension of a cyclic group by a cyclic group. ExamplesExamples of polycyclic groups include finitely generated abelian groups, finitely generated nilpotent groups, and finite solvable groups. Anatoly Maltsev proved that solvable subgroups of the integer general linear group are polycyclic; and later Louis Auslander (1967) and Swan proved the converse, that any polycyclic group is up to isomorphism a group of integer matrices.[1] The holomorph of a polycyclic group is also such a group of integer matrices.[2] Strongly polycyclic groupsA group G is said to be strongly polycyclic if it is polycyclic with the added stipulation that each Gi / Gi+1 is infinitely cyclic. Clearly, a strongly polycyclic group is polycyclic. Also, any subgroup of a strongly polycyclic group is strongly polycyclic. Polycyclic-by-finite groupsA virtually polycyclic group is a group that has a polycyclic subgroup of finite index, an example of a virtual property. Such a group necessarily has a normal polycyclic subgroup of finite index, and therefore such groups are also called polycyclic-by-finite groups. 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 residually finite. In the textbook {{harv|Scott|1964|loc=Ch 7.1}} and some papers, an M-group refers to what is now called a polycyclic-by-finite group, 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 cyclic group. These groups are particularly interesting because they are the only known examples of Noetherian group rings {{harv|Ivanov|1989}}, or group rings of finite injective dimension.{{Citation needed|date=February 2008}} Hirsch lengthThe Hirsch length or Hirsch number of a polycyclic group G is the number of infinite factors in its subnormal series. If G is a polycyclic-by-finite group, then the Hirsch length of G is the Hirsch length of a polycyclic normal subgroup H of G, where H has finite index in G. This is independent of choice of subgroup, as all such subgroups will have the same Hirsch length. See also
References{{Refimprove|date=June 2008}}
Notes1. ^Dmitriĭ Alekseevich Suprunenko, K. A. Hirsch, Matrix groups (1976), pp. 174–5; [https://books.google.com/books?id=cTtuPOj5h10C&pg=PA174 Google Books]. 2. ^{{SpringerEOM|id=p/p073560|title=Polycyclic group}} 2 : Properties of groups|Solvable groups |
| 随便看 |
|
开放百科全书收录14589846条英语、德语、日语等多语种百科知识，基本涵盖了大多数领域的百科知识，是一部内容自由、开放的电子版国际百科全书。