Groups where non-normal subgroups are close to Abelian groups

M.M. Semko; O.A. Yarova

doi:10.15421/241117

Groups where non-normal subgroups are close to Abelian groups

M.M. Semko (National University of the State Tax Service of Ukraine), https://orcid.org/0000-0003-0123-4872
O.A. Yarova (National University of the State Tax Service of Ukraine), https://orcid.org/0000-0002-0522-8368

Abstract

In this paper we consider solvable groups, each subgroup of which is either normal or has a Chernikov commutator subgroup.

Keywords

Chernikov group; solvable group

Full Text:

PDF (Українська)

References

Chernikov S.N. "Groups with given properties of systems of infinite subgroups", Ukrainian Math. J. doi:10.1007/BF01105854

Chernikov S.N. "Infinite non-Abelian groups with minimality condition for non-normal subgroups", Math. Notes, 1969; 6(1): pp. 11-18. doi:10.1007/BF01450247

Kuzennyi M.F., Semko M.M. Metahamiltonian groups and their generalizations, Kyiv, 1996. (in Ukrainian)

Kurdachenko L.A. "Locally nilpotent groups with weak minimality condition for normal subgroups", Siberian Math. J., 1984; 25(4): pp. 589-593. doi:10.1007/BF00968897

Fuchs L. Infinite abelian groups, Academic Press, 1970.

Belyaev V.V. "Locally finite groups with Chernikov Sylow $$$p$$$-subgroups", Algebra Logic, 1981; 20(7): pp. 605-619. doi:10.1007/BF01669128

Dedekind R. "Uber Gruppen, deren sämmtliche Teiler Normalteiler sind", Math. Ann., 1897; 48(4): pp. 548-561. (in German) doi:10.1007/BF01447922

Baer R. "Situation der Untergruppen, und Struktur der Gruppe", S.-B. Heidelberg Acad. Math.-Nat. Klasse, 1933; 2: pp. 12-17. (in German)

Kurdachenko L.A., Otal J., Russo A., Vincenci G. "The local structure of groups whose non-normal subgroups have finite conjugacy classes", Adv. Group Theory 2002. Proc. Intens. Bimester Dedic. R. Baer, Roma; pp. 93-110.

Kurdachenko L.A., Otal J., Russo A., Vincenci G. "Groups whose non-normal subgroups have finite conjugacy classes", Math. Proc. Royal Irish Acad., 2004; 104A(2): pp. 177-189.

Semko N.N., Yarovaya O.A. "On some generalization of metahamiltonian groups", Algebra Discr. Math., 2009; 2: pp. 16-24.

Dixon M.R. Sylow theory, formations and Fitting classes in locally finite groups, World Scientific, 1994.

Kurdachenko L.A., Otal J., Subbotin I.Ya. Artinian modules over group rings, Birkhauser, 2007.

Shmidt O.Yu. "Groups that have only one class of non-invariant subgroups", Matem. sbor., 1926; 33: pp. 161-172.

Shmidt O.Yu. "Groups with two classes of non-invariant subgroups", Works of Seminar on Group Theory, 1938; pp. 7-26.

Romalis G.M., Sesekin N.F. "On metahamiltonian groups", Matem. zap. ural univ., 1966; 5(3): pp. 45-49.

Sesekin N.F., Romalis G.M. "On metahamiltonian groups II", Matem. zap. ural univ., 1968; 6(5): pp. 50-53.

Romalis G.M., Sesekin N.F. "On metahamiltonian groups III", Matem. zap. ural univ., 1970; 7(3): pp. 195-199.

Nagrebetskii V.T. "Finite non-nilpotent groups, any non-Abelian subgroup of which is invariant", Matem. zap. ural univ., 1967; 6(1): pp. 80-88.

Makhnev A.A. "On finite metahamiltonian groups", Matem. zap. ural univ., 1976; 10(1): pp. 60-75.

Yarova O.A. "On groups that are close to metahamiltonian", Res. Math., 2009; 14: pp. 143-149.

Yarova O.A. "On groups, all proper subgroups of which are close to Abelian", Visnyk Kyiv. univ. Ser. fiz.-mat. nauk, 2008; 4: pp. 36-39.

Kargapolov M.I. "Locally finite groups that have normal systems with finite factors", Siberian Math. J., 1961; 2(6): pp. 853-873.