Abstract
We briefly review some recent works on groups with the weak minimal condition on nonnilpotent subgroups. We also study the class of groups with the weak minimal condition on non-(soluble of derived length d) subgroups.
Similar content being viewed by others
REFERENCES
S. N. Černikov, “Infinite locally soluble groups,” Mat. Sb., 7, 35–64 (1940).
S. N. Černikov, “On the theory of locally soluble groups with the minimal condition for subgroups,” Dokl. Akad. Nauk SSSR, 65, 21–24 (1949).
S. N. Černikov, “On locally solvable groups satisfying the minimality condition for subgroups,” Mat. Sb., 28(70), 119–129 (1951).
R. Baer, “Nil-Gruppen,” Math. Z., 62, 402–437 (1955).
O. J. Schmidt, “Infinite soluble groups,” Mat. Sb., 17(59), 145–162 (1945).
V. P. Šunkov, “On locally finite groups with a minimality condition for abelian subgroups,” Algebra Logika, 9, 579–615 (1970), English Transl.: Algebra Logic, 9, 350–370 (1970).
O. H. Kegel and B. A. F. Wehrfritz, “Strong finiteness conditions in locally finite groups,” Math. Z., 117, 309–324 (1970).
S. N. Černikov, “Infinite groups with certain prescribed properties of systems of infinite subgroups of them,” Dokl. Akad. Nauk SSSR, 159, 759–760 (1964).
S. N. Černikov, “Groups with prescribed properties for systems of infinite subgroups,” Ukr. Mat. Zh., 19, No. 1, 111–131 (1967).
D. I. Zaicev, “Groups which satisfy a weak minimality condition,” Dokl. Akad. Nauk SSSR, 178, 780–782 (1968), English Transl.: Sov. Math. Dokl., 9, 194–197 (1968).
D. I. Zaicev, “Groups which satisfy a weak minimality condition,” Ukr. Mat Zh., 20, No. 5, 472–482 (1968.), English Transl.: Ukr. Math. J., 20, No. 5, 408–416 (1968).
R. Baer, “Polyminimaxgruppen,” Math. Ann., 175, 1–43 (1968).
D. I. Zaicev, “Groups satisfying the weak minimal condition for non-abelian subgroups,” Ukr. Mat. Zh., 23, No. 6, 661–665 (1971), English Transl.: Ukr. Math. J., 23, No. 6, 543–546 (1971).
D. I. Zaicev, “Theory of minimax groups,” Ukr. Mat. Zh., 23, No. 6, 652–660 (1971), English Transl.: Ukr. Math. J., 23, No. 6, 536–542 (1971).
M. R. Dixon and M. J. Evans, “Groups with the minimum condition on insoluble subgroups,” Arch. Math., 72, 241–251 (1999).
M. R. Dixon, M. J. Evans, and H. Smith, “Groups with some minimal conditions on non-nilpotent subgroups,” J. Group Theory, 4, 207–215 (2001).
M. R. Dixon, M. J. Evans, and H. Smith, “Locally soluble-by-finite groups with the weak minimal condition for non-nilpotent subgroups,” J. Algebra (to appear).
A. O. Asar, “Locally nilpotent p-groups whose proper subgroups are hypercentral or nilpotent-by-Chernicov,” J. London Math. Soc., 61, 412–422 (2000).
F. Napolitani and E. Pegoraro, “On groups with nilpotent by Černikov proper subgroups,” Arch. Math., 69, 89–94 (1997).
D. I. Zaicev, “Stably solvable groups,” Izv. Akad. Nauk SSSR. Ser. Mat., 33, 765–780 (1969), English Transl.: Math. USSR-Izv., 3, 723–736 (1969).
A. N. Ostylovskii, “A weak minimality condition for nonnilpotent subgroups,” Algebra Logika, 23, 439–444 (1984), English Transl.: Algebra Logic, 23, 303–306 (1984).
H. Smith, “Groups with few non-nilpotent subgroups,” Glasgow Math. J., 39, 141–151 (1997).
D. J. S. Robinson, A Course in the Theory of Groups, Grad. Texts Math., Vol. 80, Springer, Berlin etc. (1996).
L. S. Kazarin, “Soluble product of groups,” in: F. de Giovanni and M. L. Newell (editors), Infinite Groups, Walter de Gruyter, New York (1996), pp. 111–123.
D. J. S. Robinson, Finiteness Conditions and Generalized Soluble Groups, Ergeb. Math., Vols. 1, 2, Springer, Berlin etc. (1972).
P. Hall, The Edmonton Notes on Nilpotent Groups, Queen Mary College Mathematics Notes. (1969).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Dixon, M.R., Evans, M.J. & Smith, H. Groups with Various Minimal Conditions on Subgroups. Ukrainian Mathematical Journal 54, 957–966 (2002). https://doi.org/10.1023/A:1021712220100
Issue Date:
DOI: https://doi.org/10.1023/A:1021712220100