Abstract:
An “economical” imbedding theorem is proved for at most a denumerable set of groups of finite or denumerable cardinality without involution in a group with “few” subgroups. This result is used to solve a series of problems about groups satisfying the descending chain condition for subgroups; in particular, a nondenumerable group with this condition is constructed.
Bibliography: 20 titles.
\Bibitem{Obr89}
\by V.~N.~Obraztsov
\paper An imbedding theorem for groups and its corollaries
\jour Math. USSR-Sb.
\yr 1990
\vol 66
\issue 2
\pages 541--553
\mathnet{http://mi.mathnet.ru/eng/sm2990}
\crossref{https://doi.org/10.1070/SM1990v066n02ABEH001184}
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\zmath{https://zbmath.org/?q=an:0685.20030|0698.20021}
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Linking options:
https://www.mathnet.ru/eng/sm2990
https://doi.org/10.1070/SM1990v066n02ABEH001184
https://www.mathnet.ru/eng/sm/v180/i4/p529
This publication is cited in the following 30 articles:
M. De Falco, C. Musella, G. Sabatino, “Uncountable groups with finitely many normalizers of large subgroups”, Arch. Math., 122:4 (2024), 369
Maria De Falco, Francesco de Giovanni, Carmela Musella, “Pronormality in uncountable groups”, Communications in Algebra, 2024, 1
Martina Capasso, Liliana Lancellotti, “Uncountable groups in which permutability is a transitive relation”, Journal of Group Theory, 2024
GIULIA SABATINO, “GROUPS IN WHICH ALL LARGE SUBGROUPS HAVE BOUNDED NEAR DEFECT”, Bull. Aust. Math. Soc., 107:3 (2023), 456
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Francesco de Giovanni, Marco Trombetti, “Groups whose proper subgroups are metahamiltonian-by-finite”, Rocky Mountain J. Math., 50:1 (2020)
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Francesco De Giovanni, Marco Trombetti, “Groups with restrictions on proper uncountable subgroups”, Studia Scientiarum Mathematicarum Hungarica, 56:2 (2019), 154
Francesco de Giovanni, Marco Trombetti, “The class of minimax groups is countably recognizable”, Monatsh Math, 185:1 (2018), 81
M. De Falco, M. J. Evans, F. de Giovanni, C. Musella, “Permutability in uncountable groups”, Annali di Matematica, 197:5 (2018), 1417
FRANCESCO DE GIOVANNI, MARCO TROMBETTI, “A NOTE ON GROUPS WHOSE PROPER LARGE SUBGROUPS HAVE A TRANSITIVE NORMALITY RELATION”, Bull. Aust. Math. Soc., 95:1 (2017), 38
Francesco de Giovanni, Federica Saccomanno, “A note on groups whose proper subgroups are quasihamiltonian-by-finite”, Ricerche mat, 66:2 (2017), 619
Francesco de Giovanni, Marco Trombetti, “A note on uncountable groups with modular subgroup lattice”, Arch. Math., 107:6 (2016), 581
L. A. Kurdachenko, I. Ya. Subbotin, N. A. Turbay, “On influence of formation theory concepts and ideology on infinite group theory”, Tr. In-ta matem., 21:1 (2013), 69–77
V. S. Atabekyan, “On normal subgroups in the periodic products of S. I. Adian”, Proc. Steklov Inst. Math., 274 (2011), 9–24
M. De Falco, F. de Giovanni, C. Musella, “Groups in Which Every Subgroup Is Permutable-by-Finite”, Communications in Algebra, 32:3 (2004), 1007
Dmitriy Sonkin, “CEP-Subgroups of Free Burnside Groups of Large Odd Exponents”, Communications in Algebra, 31:10 (2003), 4687
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