In this work, characterizations of a locally graded periodic group whose proper subgroups are residually nilpotent are obtained. It is shown that if such a group is minimal non-(residually nilpotent), then, under certain conditions, it has a homomorphic image which is a barely transitive group. In particular a minimal non-hypercentral group whose proper subgroups are residually nilpotent has a barely transitive homomorphic image. As an application a related question about Heineken–Mohamed-type groups is answered. Finally, a short proof and a generalization of a result on the solvability of locally graded groups is given.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Asar, A.O.: Locally nilpotent \(p\)-groups whose proper subgroups are hypercentral or nilpotent-by-Chernikov. J. Lond. Math. Soc. 61(2), 412–422 (2000)
Asar, A.O.: On infinitely generated groups whose proper subgroups are solvable. J. Algebra 399, 870–886 (2014)
Asar, A.O.: On Fitting groups whose proper subgroups are solvable. Int. J. Group Theory 2, 7–24 (2016)
Belyaev, V.V., Kuzucuoglu, M.: Barely transitive and Heineken Mohamed groups. J. Lond. Math Soc. 55(2), 261–263 (1997)
de Giovanni, F., Trombetti, M.: Infinite minimal non-hypercyclic groups. J. Algebra Appl. 14, 1550143 (2015)
Dixon, J.D., Mortimer, B.: Permutation Groups. Springer-Ferlag, New York (1996)
Hartley, B.: On the normalizer condition and barely transitive permutation groups. Algebra Log. 13, 334–340 (1974)
Heineken, H., Mohamed, I.J.: A group with trivial centre satisfying the normalizer condition. J. Algebra 10, 368–376 (1968)
Kegel, O.H., Wehrfritz, B.A.F.: Locally Finite Groups. North-Holland Amsterdam, London (1973)
Khukhro, E.I., Makarenko, NYu.: Large carracteristic subgroupss satisfying multilinear commutator identities. J. Lond. Math. Soc. 75(2), 634–646 (2007)
Möhres, W.: Torsionsgruppen, deren Untergruppen alle subnormal sind. Geom. Dedic. 31, 237–244 (1989)
Möhres, W.: Auflösbare gruppen mith endlichem exponenten, deren Untergruppen alle subnormal sind-II. Rend. Sem. Mat. Univ. Padova 81, 269–287 (1989)
Newman, M.F., Wiegold, J.: Groups with many nilpotent subgroups. Arch. der Math. 15, 241–250 (1964)
Robinson, D.J.S.: Finiteness Conditions and Generalized Solvable Groups Part I, II. Springer, New York (1972)
Robinson, D.J.S.: A Course in the Theory of Groups. Springer-Verlag, New York (1980)
Smith, H.: Groups with few non-nilpotent subgroups. Glasg. Math. J. 39, 141–151 (1997)
Zel’manov, E.I.: Solution of the restricted Burnside problem for groups of odd exponent. Math. USSR-Izv. 36, 41–60 (1991)
Zel’manov, E.I.: Solution of the restricted Burnside problem for 2-groups. Math. USSR-Sb. 72, 543–565 (1992)
The author is very grateful to the referee for a careful reading of the manuscript and for some in valuable suggestiions.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Dedicated to Professor Berhard Amberg.
About this article
Cite this article
Asar, A.O. On Minimal Non-(residually Nilpotent) Locally Graded Groups. Mediterr. J. Math. 18, 54 (2021). https://doi.org/10.1007/s00009-020-01664-7
Mathematics Subject Classification
- 20E 25
- 20E 26
- 20F 19
- 20F 50