×

On \(PC\)-hypercentral and \(CC\)-hypercentral groups. (English) Zbl 0911.20029

Let \(\mathfrak X\) be a class of groups which is \(H\) and \(R_0\)-closed (i.e. \(\mathfrak X\) is quotient closed and \(G/N_1\cap N_2\) is an \(\mathfrak X\)-group provided that \(N_1\) and \(N_2\) are normal subgroups of the group \(G\) such that \(G/N_1\) and \(G/N_2\) belong to \(\mathfrak X\)). If \(G\) is any group, the \({\mathfrak X}C\)-centre \({\mathfrak X}C(G)\) of \(G\) is the subgroup consisting of all elements \(x\) of \(G\) such that \(G/C_G(x^G)\) belongs to \(\mathfrak X\). The upper \({\mathfrak X}C\)-central series of \(G\) is then defined by the rules \({\mathfrak X}C_0(G)=\{1\}\), \({\mathfrak X}C_{\alpha+1}(G)/{\mathfrak X}C_\alpha(G)={\mathfrak X}C(G/{\mathfrak X}C_\alpha(G))\) for each ordinal \(\alpha\) and \({\mathfrak X}C_\lambda(G)=\bigcup_{\beta<\lambda}{\mathfrak X}C_\beta(G)\) when \(\lambda\) is a limit ordinal. The group \(G\) is said to be \({\mathfrak X}C\)-hypercentral if \({\mathfrak X}C_\tau(G)=G\) for some ordinal \(\tau\). A group \(G\) is called \(PC\)-hypercentral (respectively: \(CC\)-hypercentral) if it is \({\mathfrak X}C\)-hypercentral, where \(\mathfrak X\) is the class of all polycyclic-by-finite groups (respectively: of all Chernikov groups). Properties of \(PC\)-hypercentral and \(CC\)-hypercentral groups with restrictions on the chains of normal subgroups are studied in this article.

MSC:

20F24 FC-groups and their generalizations
20F14 Derived series, central series, and generalizations for groups
20F16 Solvable groups, supersolvable groups
20E15 Chains and lattices of subgroups, subnormal subgroups
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Amberg B., Ricerche di Mat pp 103– (1987)
[2] DOI: 10.1007/BF01350807 · Zbl 0183.02602 · doi:10.1007/BF01350807
[3] Baer R., Illinois J. Math 16 pp 533– (1972)
[4] DOI: 10.1017/S030500410003139X · doi:10.1017/S030500410003139X
[5] Pranciosi S., Boll. Un. Mat. Ital 4 pp 35– (1990)
[6] DOI: 10.1017/S144678870003250X · doi:10.1017/S144678870003250X
[7] Pranciosi S., Bollettino U.M.L 10 pp 653– (1996)
[8] DOI: 10.1112/plms/s3-16.1.1 · Zbl 0145.24605 · doi:10.1112/plms/s3-16.1.1
[9] Landolfi T., Ricerche di Mat pp 337– (1995)
[10] DOI: 10.1017/S2040618500033414 · Zbl 0072.25702 · doi:10.1017/S2040618500033414
[11] DOI: 10.1112/plms/s3-18.3.495 · Zbl 0157.05402 · doi:10.1112/plms/s3-18.3.495
[12] DOI: 10.1016/0021-8693(70)90121-3 · Zbl 0186.32204 · doi:10.1016/0021-8693(70)90121-3
[13] Robinson D.J.S., Finiteness Conditions and Generalized Soluble Groups 2 (1972) · Zbl 0243.20032
[14] Tomkinson M.J., FC-groups, Res. Notes in Math 96 (1984) · Zbl 0547.20031
[15] DOI: 10.1007/BF01111865 · Zbl 0181.03503 · doi:10.1007/BF01111865
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.