Classical results by Mal’cev and Šunkov show that locally nilpotent groups and locally finite groups of infinite rank must contain some Abelian subgroups of infinite rank. In recent years, many authors have studied groups in which all subgroups of infinite rank have a given property (which can be either absolute or of embedding type). Results from these researches and some new contributions to this topic are described in this paper.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
N. S. Černikov, “A theorem on groups of finite special rank,” Ukrain. Math. J., 42, 855–861 (1990).
M. De Falco, F. de Giovanni, and C. Musella, “Groups whose finite homomorphic images are metahamiltonian,” Comm. Algebra, 37, 2468–2476 (2009).
M. De Falco, F. de Giovanni, and C. Musella, “Groups whose proper subgroups of infinite rank have a transitive normality relation,” Mediterranean J. Math., 10, 1999-2006 (2013).
M. De Falco, F. de Giovanni, C. Musella, and Y. P. Sysak, “On metahamiltonian groups of infinite rank,” J. Algebra (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).
M. R. Dixon, M. J. Evans, and H. Smith, “Locally (soluble-by-finite) groups with all proper insoluble subgroups of finite rank,” Arch. Math. (Basel), 68, 100–109 (1997).
M. R. Dixon, M. J. Evans, and H. Smith, “Locally (soluble-by-finite) groups with all proper nonnilpotent subgroups of finite rank,” J. Pure Appl. Algebra, 135, 33–43 (1999).
M. R. Dixon and Y. Karatas, “Groups with all subgroups permutable or of finite rank,” Centr. Eur. J. Math., 10, 950–957 (2012).
M. J. Evans and Y. Kim, “On groups in which every subgroup of infinite rank is subnormal of bounded defect,” Comm. Algebra, 32, 2547–2557 (2004).
B. Hartley, “Uncountable artinian modules and uncountable soluble groups satisfying Min-n,” Proc. London Math. Soc., 35, 55–75 (1977).
L. A. Kurdachenko and H. Smith, “Groups in which all subgroups of infinite rank are subnormal,” Glasgow Math. J., 46, 83–89 (2004).
A. I. Mal’cev, “On certain classes of infinite soluble groups,” Mat. Sb., 28, 567–588 (1951); Amer. Math Soc. Transl., 2, 1–21 (1956).
D. McDougall, “Soluble groups with the minimum condition for normal subgroups,” Math. Z., 118, 157–167 (1970).
Y. I. Merzljakov, “Locally soluble groups of finite rank,” Algebra Logic, 8, 686–690 (1969).
W. Möhres, “Aufl¨osbarkeit, von Gruppen, deren Untergruppen alle subnormal sind,” Arch. Math. (Basel), 54, 232–235 (1990).
B. H. Neumann, “Groups with finite classes of conjugate subgroups,” Math. Z., 63, 76–96 (1955).
D. J. S. Robinson, “Groups in which normality is a transitive relation,” Proc. Cambridge Philos. Soc., 68, 21–38 (1964).
D. J. S. Robinson, Finiteness Conditions and Generalized Soluble Groups, Springer, Berlin (1972).
G. M. Romalis ans N. F. Sesekin, “Metahamiltonian groups,” Ural. Gos. Univ. Mat. Zap., 5, 101–106 (1966).
G. M. Romalis and N. F. Sesekin, “Metahamiltonian groups. II,” Ural. Gos. Univ. Mat. Zap., 6, 52–58 (1968).
G. M. Romalis and N. F. Sesekin, “Metahamiltonian groups. III,” Ural. Gos. Univ. Mat. Zap., 7, 195–199 (1969/70).
J. E. Roseblade, “On groups in which every subgroup is subnormal,” J. Algebra, 2, 402–412 (1965).
R. Schmidt, Subgroup Lattices of Groups, de Gruyter, Berlin (1994).
N. N. Semko and S. N. Kuchmenko, “Groups with almost normal subgroups of infinite rank,” Ukrain. Math. J., 57, 621–639 (2005).
V. P. Šunkov, “On locally finite groups of finite rank,” Algebra Logic, 10, 127–142 (1971).
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Nikolai Vavilov on the occasion of his 60th birthday
Published in Zapiski Nauchnykh Seminarov POMI, Vol. 414, 2013, pp. 31–39.
Rights and permissions
About this article
Cite this article
de Giovanni, F. Infinite Groups with Rank Restrictions on Subgroups. J Math Sci 199, 261–265 (2014). https://doi.org/10.1007/s10958-014-1854-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-014-1854-7