Abstract
A group G is called a T-group if all its subnormal subgroups are normal, and G is a \({\bar{T}}\) -group if every subgroup of G has the property T. It is proved here that if G is a locally soluble group whose proper subgroups of infinite rank have the T-property, then either G is a \({\bar{T}}\) -group or it has finite rank.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
B. Amberg, S. Franciosi and F. de Giovanni, Products of Groups, Clarendon Press, 1992.
De Falco M., de Giovanni F.: Groups with many subgroups having a transitive normality relation. Bol. Soc. Brasil. Mat. 31, 73–80 (2000)
M. De Falco, F. de Giovanni, C. Musella ans Y.P. Sysak, On metahamiltonian groups of infinite rank, To appear.
M. De Falco, F. de Giovanni, C. Musella and Y. P. Sysak, Groups of infinite rank in which normality is a transitive relation, Glasgow Math. J., to appear.
M. De Falco, F. de Giovanni, C. Musella and N. Trabelsi, Groups with restrictions on subgroups of infinite rank, Rev. Mat. Iberoamericana, to appear.
M. De Falco, F. de Giovanni, C. Musella and N. Trabelsi, Groups whose proper subgroups of infinite rank have finite conjugacy classes, Bull. Austral. Math. Soc., to appear.
Dixon M. R., Evans M. J., Smith H.: Locally soluble-by-finite groups of finite rank. J. Algebra 182, 756–769 (1996)
Dixon M. R., Evans M. J., Smith H.: Locally (soluble-by-finite) groups with all proper non-nilpotent subgroups of finite rank. J. Pure Appl. Algebra 135, 33–43 (1999)
Dixon M. R., Evans M. J., Smith H.: Groups with all proper subgroups (finite rank)-by-nilpotent. Arch. Math. (Basel) 72, 321–327 (1999)
Dixon M.R., Karatas Z.Y.: Groups with all subgroups permutable or of finite rank. Centr. Eur. J. Math. 10, 950–957 (2012)
Evans M.J., Kim Y.: On groups in which every subgroup of infinite rank is subnormal of bounded defect. Comm. Algebra 32, 2547–2557 (2004)
Gaschütz W.: Gruppen in denen das Normalteilersein transitiv ist. J. Reine Angew. Math. 198, 87–92 (1957)
Robinson D. J. S.: Groups in which normality is a transitive relation. Proc. Cambridge Philos. Soc. 68, 21–38 (1964)
Robinson D. J. S.: Groups which are minimal with respect to normality being intransitive. Pacific J. Math. 31, 777–785 (1969)
D. J. S. Robinson, Finiteness Conditions and Generalized Soluble Groups, Springer, 1972.
Robinson D. J. S.: Splitting theorems for infinite groups. Symposia Mathematica 17, 441–470 (1973)
D. J. S. Robinson, A Course in the Theory of Groups, 2nd edition, Springer, 1996.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
De Falco, M., de Giovanni, F. & Musella, C. Groups whose Proper Subgroups of Infinite Rank Have a Transitive Normality Relation. Mediterr. J. Math. 10, 1999–2006 (2013). https://doi.org/10.1007/s00009-013-0321-x
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00009-013-0321-x