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ON GROUPS WITH ALL SUBGROUPS SUBNORMAL OR SOLUBLE OF BOUNDED DERIVED LENGTH

Published online by Cambridge University Press:  13 August 2013

KIVANÇ ERSOY ,
ANTONIO TORTORA  and
MARIA TOTA
KIVANÇ ERSOY
Affiliation:
Department of Mathematics, Mimar Sinan Fine Arts Universityİstanbul 34427, Turkey e-mail: kivanc.ersoy@msgsu.edu.tr
ANTONIO TORTORA
Affiliation:
Dipartimento di Matematica, Università di Salerno Via Giovanni Paolo II, 132 - Fisciano (SA) 84084, Italy e-mail: antortora@unisa.it, mtota@unisa.it
MARIA TOTA
Affiliation:
Dipartimento di Matematica, Università di Salerno Via Giovanni Paolo II, 132 - Fisciano (SA) 84084, Italy e-mail: antortora@unisa.it, mtota@unisa.it
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Abstract

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In this paper we deal with locally graded groups whose subgroups are either subnormal or soluble of bounded derived length, say d. In particular, we prove that every locally (soluble-by-finite) group with this property is either soluble or an extension of a soluble group of derived length at most d by a finite group, which fits between a minimal simple group and its automorphism group. We also classify all the finite non-abelian simple groups whose proper subgroups are metabelian.

Keywords

20F1920E32
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 56 , Issue 1 , January 2014 , pp. 221 - 227
Copyright
Copyright © Glasgow Mathematical Journal Trust 2013 

References

REFERENCES

1.Aríkan, A., Sezer, S. and Smith, H., On locally finite minimal non-solvable groups, Cent. Eur. J. Math. 8 (2010), 266–273.CrossRefGoogle Scholar
2.Bloom, D. M., The subgroups of PSL(3, q) for odd q, Trans. Am. Math. Soc. 127 (1967), 150–178.Google Scholar
3.Casolo, C., Groups in which all subgroups are subnormal, Rend. Accad. Naz. Sci. XL Mem. Mat. 10 (5) (1986), 247–249.Google Scholar
4.Casolo, C., Torsion-free groups in which every subgroup is subnormal, Rend. Circ. Mat. Palermo 50 (2) (2001), 321–324.Google Scholar
5.Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A. and Wilson, R. A., Atlas of finite groups (Oxford University Press, Oxford, UK, 1985).Google Scholar
6.Dixon, M. R. and Evans, M. J., Groups with the minimum condition on insoluble subgroups, Arch. Math. 72 (1999), 241–251.Google Scholar
7.Dixon, M. R., Evans, M. J. and Smith, H., Groups with proper subgroups of certain types, Ischia Group Theory 2006 (World Scientific Publishing, Hackensack, NJ, 2007), 73–82.Google Scholar
8.Franciosi, S., de Giovanni, F. and Newell, M. L., Groups with polycyclic non-normal subgroups, Algebra Colloq. 7 (2000), 33–42.CrossRefGoogle Scholar
9.Lennox, J. C. and Robinson, D. J. S., The theory of infinite soluble groups (Oxford University Press, Oxford, UK, 2004).CrossRefGoogle Scholar
10.Longobardi, P., Maj, M. and Smith, H., A note on locally graded groups, Rend. Sem. Mat. Univ. Padova 94 (1995), 275–277.Google Scholar
11.Möhres, W., Torsionsfreie Gruppen, deren Untergruppen alle subnormal sind, Math. Ann. 284 (1989), 245–249.CrossRefGoogle Scholar
12.Möhres, W., Auflösbarkeit von Gruppen, deren Untergruppen alle subnormal sind, Arch. Math. 54 (1990), 232–235.CrossRefGoogle Scholar
13.Olshanskii, A. Yu., Groups of bounded period with subgroups of prime order, Algebra Logic 21 (1982), 369–418.Google Scholar
14.Robinson, D. J. S., Finiteness conditions and generalized soluble groups, Part 1 and Part 2 (Springer-Verlag, Berlin, Germany, 1972).CrossRefGoogle Scholar
15.Roseblade, J. E., On groups in which every subgroup is subnormal, J. Algebra 2 (1965), 402–412.Google Scholar
16.Smith, H., Torsion-free groups with all subgroups subnormal, Arch. Math. 76 (2001), 1–6.CrossRefGoogle Scholar
17.Smith, H., Torsion-free groups with all non-nilpotent subgroups subnormal, Topics in Infinite Groups, 297–308, Quad. Mat. 8 (Dept. Math., Seconda Univ. Napoli, Caserta, 2001).Google Scholar
18.Smith, H., Groups with all non-nilpotent subgroups subnormal, topics in infinite groups, Quaderni di Matematica, 8 (2001), 309–326 (Dept. of Math., Seconda Univ. Napoli, Caserta).Google Scholar
19.Smith, H., Groups with all subgroups subnormal or nilpotent-by-Chernikov, Rend. Sem. Mat. Univ. Padova 126 (2011), 245–253.Google Scholar
20.Suzuki, M., On a class of doubly transitive groups, Ann. Math. 75 (1962), 105–145.Google Scholar
21.Suzuki, M., Group theory I (Springer-Verlag, Berlin, Germany, 1982).Google Scholar
22.Thompson, J. G., Nonsolvable finite groups all of whose local subgroups are solvable, Bull. Amer. Math. Soc. 74 (1968), 383–437.Google Scholar
23.Wilson, R. A., The finite simple groups, Graduate Texts in Mathematics, No. 251 (Springer-Verlag, London, 2009).Google Scholar