A note on groups with many locally supersoluble subgroups

Document Type: Ischia Group Theory 2014

Authors

Dipartimento di Matematica e Applicazioni - University of Napoli Federico II

Abstract

It is proved here that if $G$ is a locally graded group satisfying the minimal condition on subgroups which are not locally supersoluble‎, ‎then $G$ is either locally supersoluble or a Cernikov group‎. ‎The same conclusion holds for locally finite groups satisfying the weak minimal condition on non-(locally supersoluble) subgroups‎. ‎As a consequence‎, ‎it is shown that any infinite locally graded group whose non-(locally supersoluble) subgroups lie into finitely many conjugacy classes must be locally supersoluble.

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Main Subjects


B. Amberg, S. Franciosi and F. de Giovanni (1992). Products of Groups. Clarendon Press, Oxford.
A. O. Asar (2000). Locally nilpotent p-groups whose proper subgroups are hypercentral or nilp otent-by-Chernikov. J. London Math. Soc. (2). 61, 412-422
M. De Falco (1999). Groups satisfying the minimal condition on non-supersoluble subgroups. Ricerche Mat.. 48, 353-360
K. Doerk (1966). Minimal nicht uberauosbare, endliche Grupp en. Math. Z.. 91, 198-205
S. Franciosi, F. de Giovanni and Y. P. Sysak (1999). Groups with many F C-subgroups. J. Algebra. 218, 165-182
F. de Giovanni and M. Trombetti Infinite minimal non-hypercyclic groups. J. Algebra Appl., to app ear DOI: 10.1142/S0219498815501431.
O. H. Kegel and B. A. F. Wehrfritz (1973). Local ly Finite Groups. North-Holland Mathematical Library, North-Holland Publishing Co., Amsterdam-London, American Elsevier Publishing Co., Inc., New York. 3
P. B. Kleidman and R. A. Wilson (1987). A characterization of some lo cally nite simple groups of Lie typ e. Arch. Math. (Basel). 48, 10-14
F. Napolitani and E. Pegoraro (1997). On groups with nilpotent-by- Cernikov proper subgroups. Arch. Math. (Basel). 69, 89-94
A. Y. Ol'shanski (1991). Geometry of Defining Relations in Groups. Kluwer, Dordrecht.
D. J. S. Robinson (1972). Finiteness Conditions and Generalized Soluble Groups. Springer, Berlin.
B. A. F. Wehrfritz (1973). Infinite Linear Groups. Springer-Verlag, New York-Heidelb erg.
D. I. Zaicev (1968). Groups satisfying the weak minimal condition. Ukrain. Math. J.. 20, 408-416