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SEMIPERMUTABILITY IN GENERALISED SOLUBLE GROUPS

Part of: Structure and classification of infinite or finite groups Representation theory of groups

Published online by Cambridge University Press:  02 November 2016

A. BALLESTER-BOLINCHES ,
J. C. BEIDLEMAN  and
R. IALENTI
A. BALLESTER-BOLINCHES*
Affiliation:
Departament de Matemàtiques, Universitat de València, Dr. Moliner, 50, 46100 Burjassot, València, Spain email Adolfo.Ballester@uv.es
J. C. BEIDLEMAN
Affiliation:
Department of Mathematics, University of Kentucky, Lexington, KY 40506-0027, USA email james.beidleman@uky.edu
R. IALENTI
Affiliation:
Dipartimento di Matematica e Applicazioni, Università di Napoli Federico II, Complesso Universitario Monte S. Angelo, Via Cintia, I-80126 Napoli, Italy email roberto.ialenti@unina.it
*
✉Adolfo.Ballester@uv.es
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Abstract

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Some classes of finitely generated hyperabelian groups defined in terms of semipermutability and S-semipermutability are studied in the paper. The classification of finitely generated hyperabelian groups all of whose finite quotients are PST-groups recently obtained by Robinson is behind our results. An alternative proof of such a classification is also included in the paper.

Keywords

generalised soluble groupspermutabilityS-semipermutability

MSC classification

Primary: 20E15: Chains and lattices of subgroups, subnormal subgroups
Secondary: 20D40: Products of subgroups
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 95 , Issue 2 , April 2017 , pp. 219 - 227
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

Footnotes

The first author has been supported by the grant MTM2014-54707-C3-1-P from the Ministerio de Economía y Competitividad, Spain, and FEDER, European Union. He has also been supported by a project from the National Natural Science Foundation of China (NSFC, No. 11271085) and a project of the Natural Science Foundation of Guangdong Province (No. 2015A030313791).

References

Ballester-Bolinches, A., Beidleman, J. C., Esteban-Romero, R. and Ragland, M. F., ‘On a class of supersoluble groups’, Bull. Aust. Math. Soc. 90 (2014), 220–226.Google Scholar
Ballester-Bolinches, A., Beildleman, J. C., Feldman, A. D., Heineken, H. and Ragland, M. F., ‘Finite solvable groups in which semi-normality is a transitive relation’, Beiträge Algebra Geom. 54(2) (2013), 549–558.Google Scholar
Ballester-Bolinches, A., Esteban-Romero, R. and Asaad, M., Products of Finite Groups, de Gruyter Expositions in Mathematics, 53 (Walter de Gruyter, Berlin, 2010).Google Scholar
Ballester-Bolinches, A., Kurdachenko, L. A., Otal, J. and Pedraza, T., ‘Infinite groups with Sylow permutable subgroups’, Ann. Mat. Pura Appl. (4) 189(4) (2010), 553–565.CrossRefGoogle Scholar
Ballester-Bolinches, A., Kurdachenko, L. A. and Pedraza, T., ‘On periodic radical groups in which permutability is a transitive relation’, J. Pure Appl. Algebra 6 (2007), 527–551.Google Scholar
Beidleman, J. C. and Ragland, M. F., ‘Subnormal, permutable, and embedded subgroups in finite groups’, Cent. Eur. J. Math. 9(4) (2011), 915–921.Google Scholar
Beidleman, J. C. and Ragland, M. F., ‘Groups with maximal subgroups of Sylow subgroups satisfying certain permutability conditions’, Southeast Asian Bull. Math. 38(2) (2014), 183–190.Google Scholar
Chen, X. Y. and Guo, W. B., ‘Finite groups in which SS-permutability is a transitive relation’, Acta Math. Hungar. 143(2) (2014), 466–479.CrossRefGoogle Scholar
Doerk, K. and Hawkes, T., Finite Soluble Groups, de Gruyter Expositions in Mathematics, 4 (Walter de Gruyter, Berlin, New York, 1992).CrossRefGoogle Scholar
Heineken, H. and Beidleman, J. C., ‘T-groups, polycyclic groups, and finite quotients’, Arch. Math. 103 (2014), 21–26.Google Scholar
Kargapolov, M. I. and Merzlyakov, Ju. I., Fundamentals of the Theory of Groups, Graduate Texts in Mathematics, 62 (Springer, New York, Heidelberg, Berlin, 1979), translated from the 2nd Russian edn by R. G. Burns.Google Scholar
Kegel, O. H., ‘Sylow-Gruppen und Subnormalteiler endlicher Gruppen’, Math. Z. 78 (1962), 205–221.CrossRefGoogle Scholar
Menegazzo, F., ‘Gruppi nei quali la relazione di quasi-normalità è transitiva’, Rend. Semin. Mat. Univ. Padova 40 (1968), 347–361.Google Scholar
Menegazzo, F., ‘Gruppi nei quali la relazione di quasi-normalità è transitiva’, Rend. Semin. Mat. Univ. Padova 42 (1969), 389–399.Google Scholar
Robinson, D. J. S., ‘Sylow permutability in generalized soluble groups’, J. Group Theory (2016), ISSN (Online) 1435–4446, doi:10.1515/jgth-2016-0030.Google Scholar
Robinson, D. J. S., ‘Groups in which normality is a transitive relation’, Math. Proc. Cambridge Philos. Soc. 60 (1964), 21–38.CrossRefGoogle Scholar
Robinson, D. J. S., Finiteness Conditions and Generalized Soluble Groups (Springer, Berlin, 1972).Google Scholar
Robinson, D. J. S., A Course in the Theory of Groups (Springer, New York, 1982).Google Scholar
Robinson, D. J. S., ‘Sylow permutability in locally finite groups’, Ric. Mat. 59(2) (2010), 313–318.Google Scholar
Stonehewer, S. E., ‘Permutable subgroups of infinite groups’, Math. Z. 125 (1972), 1–16.Google Scholar
Wang, L., Li, Y. and Wang, Y., ‘Finite groups in which (S-)semipermutability is a transitive relation’, Int. J. Algebra 2(1–4) (2008), 143–152; Corrigendum in Int. J. Algebra 6(13–16) (2012), 727–728.Google Scholar