×

On Černikov-by-nilpotent groups. (English) Zbl 1162.20023

Let \(\Omega\) be a class of groups. Define \((\Omega,\infty)\) to be the class of all groups in which every infinite subset contains two distinct elements which generate an \(\Omega\)-group. In the paper under review, the author studies finitely generated solvable groups in the class \((\Omega,\infty)\) whenever \(\Omega\) is either the class of Chernikov groups (denoted by \(\check {C}\)), or the class of Chernikov-by-nilpotent groups (denoted by \(\check {C}N\)). The class of finite groups and finite-by-nilpotent groups are denoted by \(F\) and \(FN\), respectively.
The author proves the following:
Theorem 1. Let \(G\) be a finitely generated solvable group. Then \(G\in (\check {C}, \infty)\) if and only if \(G\in F\).
Theorem 2. Let \(G\) be a finitely generated solvable group. Then \(G\in (\check {C}N, \infty)\) if and only if \(G\in FN\).

MSC:

20F19 Generalizations of solvable and nilpotent groups
20E25 Local properties of groups
20F05 Generators, relations, and presentations of groups
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.1080/00927879908826779 · Zbl 0942.20014 · doi:10.1080/00927879908826779
[2] DOI: 10.1017/S0305004100031662 · doi:10.1017/S0305004100031662
[3] DOI: 10.1017/S1446788700015093 · Zbl 0273.20034 · doi:10.1017/S1446788700015093
[4] DOI: 10.1017/S1446788700024253 · doi:10.1017/S1446788700024253
[5] DOI: 10.1017/S1446788700019303 · doi:10.1017/S1446788700019303
[6] DOI: 10.1007/978-3-662-07241-7 · doi:10.1007/978-3-662-07241-7
[7] DOI: 10.1007/978-1-4684-0128-8 · doi:10.1007/978-1-4684-0128-8
[8] DOI: 10.1017/S0004972700021997 · Zbl 0959.20034 · doi:10.1017/S0004972700021997
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.