Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-26T03:43:04.235Z Has data issue: false hasContentIssue false
  • English
  • Français

ON GROUPS WHOSE SUBGROUPS ARE EITHER MODULAR OR CONTRANORMAL

Part of: Structure and classification of infinite or finite groups Special aspects of infinite or finite groups

Published online by Cambridge University Press:  18 June 2021

FAUSTO DE MARI Open the ORCID record for FAUSTO DE MARI [Opens in a new window]
FAUSTO DE MARI*
Affiliation:
Università degli Studi di Napoli Federico II, 80126, Naples, Italy
*
e-mail: zhan@math.ecnu.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

A subgroup H of a group G is said to be contranormal in G if the normal closure of H in G is equal to G. In this paper, we consider groups whose nonmodular subgroups (of infinite rank) are contranormal.

Keywords

locally soluble groupcontranormal subgroupmodular subgrouppermutable subgroupinfinite rank

MSC classification

Primary: 20E15: Chains and lattices of subgroups, subnormal subgroups
Secondary: 20F19: Generalizations of solvable and nilpotent groups
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 105 , Issue 2 , April 2022 , pp. 286 - 295
Check for updates
Copyright
© 2021 Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Casolo, C., ‘Groups with finite conjugacy classes of subnormal subgroups’, Rend. Semin. Mat. Univ. Padova 81 (1989), 107–149.Google Scholar
Cutolo, G., ‘A normalizer condition on systems of subgroups’, Boll. Unione Mat. Ital. A (7) 3 (1989), 215–223.Google Scholar
Cutolo, G. and Leone, A., ‘Permutable subgroups and the Maier–Schmid theorem for nilpotent-by-finite groups’, J. Algebra 546 (2020), 723–733.10.1016/j.jalgebra.2019.11.008CrossRefGoogle Scholar
Dardano, U. and De Mari, F., ‘On groups with all proper subgroups finite-by-abelian-by-finite’, Arch. Math. (Basel) (to appear). Published online (17 February 2021).Google Scholar
Dardano, U. and De Mari, F., ‘On groups in which subnormal subgroups of infinite rank are commensurable with some normal subgroup’, Int. J. Group Theory (to appear). Published online (12 April 2021).10.1515/jgth-2020-0076CrossRefGoogle Scholar
De Falco, M., de Giovanni, F. and Musella, C., ‘A note on groups of infinite rank with modular subgroup lattice’, Monatsh. Math. 176 (2015), 81–86.CrossRefGoogle Scholar
De Falco, M. and Musella, C., ‘A normalizer condition for modular subgroups’, in: Advances in Group Theory, Napoli, 2002 (eds. De Giovanni, F. and Newell, M. L.) (Aracne, Rome, 2002), 163–172.Google Scholar
De Luca, A. V. and di Grazia, G., ‘Groups of infinite rank with normality conditions on subgroups with small normal closure’, Bull. Aust. Math. Soc. 94 (2016), 48–53.10.1017/S0004972715001355CrossRefGoogle Scholar
De Mari, F., ‘Groups satisfying weak chain conditions on non-modular subgroups’, Comm. Algebra 46 (2018), 1709–1715.CrossRefGoogle Scholar
De Mari, F., ‘Groups with restricted non-permutable subgroups’, J. Algebra Appl. 20(4) (2021), 2150062.10.1142/S0219498821500626CrossRefGoogle Scholar
De Mari, F., ‘Groups with all subgroups either modular or soluble of finite rank’, J. Algebra 570 (2021), 416–436.10.1016/j.jalgebra.2020.12.005CrossRefGoogle Scholar
De Mari, F., ‘Groups with many modular or self-normalizing subgroups’, Comm. Algebra 49(6) (2021), 2356–2369.10.1080/00927872.2020.1870999CrossRefGoogle Scholar
Fuchs, L., Abelian Groups (Springer, Cham, 2015).CrossRefGoogle Scholar
Giordano, G., ‘Gruppi con normalizzatori estremali’, Matematiche (Catania) 26 (1971), 291–296.Google Scholar
Hall, P., ‘Some sufficient conditions for a group to be nilpotent’, Illinois J. Math. 2 (1958), 787–801.10.1215/ijm/1255448649CrossRefGoogle Scholar
Kurdachenko, L. A., Longobardi, P. and Maj, M., ‘On the structure of some locally nilpotent groups without contranormal subgroups’, Preprint, 2020, arXiv:2006.02345v1.CrossRefGoogle Scholar
Kurdachenko, L. A., Pypka, A. A. and Subbotin, I. Y., ‘On the structure of groups whose non-normal subgroups are core-free’, Mediterr. J. Math. 16(6) (2019), 136.CrossRefGoogle Scholar
Kurdachenko, L. A., Pypka, A. A. and Subbotin, I. Y., ‘On groups with normal, contranormal or core-free subgroups’, Adv. Group Theory Appl. 10 (2020), 83–125.Google Scholar
Robinson, D. J. S., ‘Groups in which normality is a transitive relation’, Math. Proc. Camb. Phil. Soc. 60 (1964), 21–38.CrossRefGoogle Scholar
Robinson, D. J. S., Finiteness Conditions and Generalized Soluble Groups (Springer, Berlin, 1972).CrossRefGoogle Scholar
Schmidt, R., Subgroup Lattices of Groups (de Gruyter, Berlin, 1994).CrossRefGoogle Scholar
Subbotin, I. Y., ‘Groups with alternatively normal subgroups’, Izv. Vyssh. Uchebn. Zaved. Mat. 3 (1992), 86–88.Google Scholar
Zacher, G., ‘Una relazione di normalità sul reticolo dei sottogruppi di un gruppo’, Ann. Mat. Pura Appl. (4) 131 (1982), 57–73.CrossRefGoogle Scholar