On groups that are nearly metahamiltonian

O.A. Yarova

doi:10.15421/240922

On groups that are nearly metahamiltonian

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

Abstract

We consider groups whose subgroups are either normal or have Chernikov commutant. It was proved that if such group G has a subgroup H of finite index and its commutant is a Chernikov group, then commutant of group G is a Chernikov group.

Full Text:

PDF (Русский)

References

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)

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

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

Chernikov S.N. "Groups with given properties of systems of infinite subgroups", Ukrainian Math. J., 1967; 19(6): pp. 111-131. (in Russian) doi:10.1007/BF01105854

Romalis G.M., Sesekin N.F. "On metahamiltonian groups I", Matem. zap. Ural un-ta, 1966; 5(3): pp. 45-49. (in Russian)

Sesekin N.F., Romalis G.M. "On metahamiltonian groups II", Matem. zap. Ural un-ta, 1968; 6(5): pp. 50-53. (in Russian)

Romalis G.M., Sesekin N.F. "On metahamiltonian groups III", Matem. zap. Ural un-ta, 1970; 7(3): pp. 195-199. (in Russian)

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

Makhnev A.A. "On finite metahamiltonian groups", Matem. zap. Ural un-ta, 1976; 10(1): pp. 60-75. (in Russian)

Kuzennyi M.F., Semko M.M. "Structure of solvable non-nilpotent metahamiltonian groups", Matem. zametki, 1983; 34(2): pp. 179-188. (in Russian) doi:10.1007/BF01141770

Kuzennyi M.F., Semko M.M. "Structure of solvable metahamiltonian groups", Dopovidi AN USSR. Ser. A, 1985; 2: pp. 6-9. (in Ukrainian)

Kuzennyi M.F., Semko M.M. "On structure of non-periodic metahamiltonian groups", Izv. vuzov. Matematika, 1986; 11: pp. 32-40. (in Russian)

Kuzennyi M.F., Semko M.M. "Structure of periodic metabelian metahamiltonian groups with non-elementary commutant", Ukrainian Math. J., 1987; 39(2): pp. 180-185. (in Russian) doi:10.1007/BF01057495

Kuzennyi M.F., Semko M.M. "On structure of periodic nonabelian metahamiltonian groups with an elementary commutant of rank three", Ukrainian Math. J., 1989; 41(2): pp. 170-176. (in Russian) doi:10.1007/BF01060379

Kuzennyi M.F., Semko M.M. "On metahamiltonian groups with elementary commutant of rank two", Ukrainian Math. J., 1990; 42(2): pp. 168-175. (in Russian) doi:10.1007/BF01071007

Kuzennyi M.F., Semko M.M. "Structure of periodic metabelian metahamiltonian groups with elementary commutant of rank two", Ukrainian Math. J., 1987; 39(6): pp. 743-750. (in Russian) doi:10.1007/BF01057181

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.

Fuchs L. Infinite abelian groups, Academic Press, New York, 1970; Vol. 1.

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

Tomkinson M.J. FC-groups, Pitman: Boston, 1984.