Geodesic
Groups satisfying certain rank conditions
Algebra and discrete mathematics, Tome 22 (2016) no. 2, pp. 184-200.

Voir la notice de l'article provenant de la source Math-Net.Ru

This is a survey of a number of recent results concerned with groups whose subgroups satisfy certain rank conditions.
Keywords: finite rank, torsion-free rank, section $p$-rank. 
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Martyn R. Dixon; Leonid A. Kurdachenko; Aleksandr A. Pypka; Igor Ya. Subbotin. Groups satisfying certain rank conditions. Algebra and discrete mathematics, Tome 22 (2016) no. 2, pp. 184-200. http://geodesic.mathdoc.fr/item/ADM_2016_22_2_a2/

[1] A. Arikan, N. Trabelsi, “On minimal non-Baer-groups”, Comm. Algebra, 39:7 (2011), 2489–2497 | DOI | MR | Zbl

[2] A. O. Asar, “Locally nilpotent $p$-groups whose proper subgroups are hypercentral or nilpotent-by-Chernikov”, J. London Math. Soc., 61:2 (2000), 412–422 | DOI | MR | Zbl

[3] A. O. Asar, “On infinitely generated groups whose proper subgroups are solvable”, J. Algebra, 399 (2014), 870–886 | DOI | MR | Zbl

[4] R. Baer, H. Heineken, “Radical groups of finite abelian subgroup rank”, Illinois J. Math., 16 (1972), 533–580 | MR | Zbl

[5] A. Ballester-Bolinches, S. Camp-Mora, L. A. Kurdachenko, J. Otal, “Extension of a Schur theorem to groups with a central factor with a bounded section rank”, J. Algebra, 393 (2013), 1–15 | DOI | MR | Zbl

[6] V. V. Belyaev, “Locally finite groups with Chernikov Sylow $p$-subgroups”, Algebra Logic, 20:6 (1981), 393–402 | DOI | MR

[7] B. Bruno, “Groups whose proper subgroups contain a nilpotent subgroup of finite index”, Boll. Un. Mat. Ital. D, 3:1 (1984), 179–188 | MR | Zbl

[8] B. Bruno, “On groups with ‘abelian by finite’ proper subgroups”, Boll. Un. Mat. Ital. B, 3:3 (1984), 797–807 | MR | Zbl

[9] B. Bruno, “On $p$-groups with nilpotent-by-finite proper subgroups”, Boll. Un. Mat. Ital. A, 3 (1989), 45–51 | MR | Zbl

[10] B. Bruno, R. E. Phillips, “A note on groups with nilpotent-by-finite proper subgroups”, Arch. Math., 65:5 (1995), 369–374 | DOI | MR | Zbl

[11] N. S. Chernikov, “A theorem on groups of finite special rank”, Ukrainian Math. J., 42:7 (1990), 855–861 | DOI | MR | Zbl

[12] F. de Giovanni, F. Saccomanno, “A note on groups of infinite rank whose proper subgroups are abelian-by-finite”, Colloq. Math., 137 (2014), 165–170 | DOI | MR | Zbl

[13] F. de Giovanni, M. Trombetti, “Groups whose proper subgroups of infinite rank have polycyclic conjugacy classes”, Algebra Colloq., 22:2 (2015), 181–188 | DOI | MR | Zbl

[14] R. Dedekind, “Über Gruppen, deren sämtliche Teiler Normalteiler sind”, Math. Ann., 48 (1897), 548–561 | DOI | MR | Zbl

[15] M. R. Dixon, M. J. Evans, H. Smith, “Locally (soluble-by-finite) groups of finite rank”, J. Algebra, 182:3 (1996), 756–769 | DOI | MR | Zbl

[16] M. R. Dixon, M. J. Evans, H. Smith, “Locally (soluble-by-finite) groups with all proper insoluble subgroups of finite rank”, Arch. Math., 68:2 (1997), 100–109 | DOI | MR | Zbl

[17] M. R. Dixon, M. J. Evans, H. Smith, “On groups that are residually of finite rank”, Israel J. Math., 107:1 (1998), 1–16 | DOI | MR | Zbl

[18] M. R. Dixon, M. J. Evans, H. Smith, “Groups with all proper subgroups (finite rank)-by-nilpotent”, Arch. Math., 72:5 (1999), 321–327 | DOI | MR | Zbl

[19] M. R. Dixon, M. J. Evans, H. Smith, “Locally (soluble-by-finite) groups with all proper non-nilpotent subgroups of finite rank”, J. Pure Appl. Algebra, 135:1 (1999), 33–43 | DOI | MR | Zbl

[20] M. R. Dixon, M. J. Evans, H. Smith, “On groups with rank restrictions on subgroups”, Groups St. Andrews 1997 in Bath, v. I, London Math. Soc. Lecture Note, 260, 1999, 237–247 | MR | Zbl

[21] M. R. Dixon, M. J. Evans, H. Smith, “A Tits alternative for groups that are residually of bounded rank”, Israel J. Math., 109:1 (1999), 53–59 | DOI | MR | Zbl

[22] M. R. Dixon, M. J. Evans, H. Smith, “Groups with some minimal conditions on non-nilpotent subgroups”, J. Group Theory, 4:2 (2001), 207–215 | DOI | MR | Zbl

[23] M. R. Dixon, M. J. Evans, H. Smith, “Groups with all proper subgroups soluble-by-finite rank”, J. Algebra, 289:1 (2005), 135–147 | DOI | MR | Zbl

[24] M. R. Dixon, Y. Karatas, “Groups with all subgroups permutable or of finite rank”, Cent. Eur. J. Math., 10:3 (2012), 950–957 | DOI | MR | Zbl

[25] M. R. Dixon, L. A. Kurdachenko, N. V. Polyakov, “Locally generalized radical groups satisfying certain rank conditions”, Ricerche Mat., 56 (2007), 43–59 | DOI | MR | Zbl

[26] M. R. Dixon, L. A. Kurdachenko, I. Ya. Subbotin, “On various rank conditions in infinite groups”, Algebra Discrete Math., 6:4 (2007), 23–44 | MR

[27] M. J. Evans, Y. Kim, “On groups in which every subgroup of infinite rank is subnormal of bounded defect”, Comm. Algebra, 32:7 (2004), 2547–2557 | DOI | MR | Zbl

[28] M. de Falco, F. de Giovanni, C. Musella, “Groups whose proper subgroups of infinite rank have a transitive normality relation”, Mediterr. J. Math., 10:4 (2013), 1999–2006 | DOI | MR | Zbl

[29] M. de Falco, F. de Giovanni, C. Musella, “Groups with finitely many conjugacy classes of non-normal subgroups of infinite rank”, Colloq. Math., 131:2 (2013), 233–239 | DOI | MR | Zbl

[30] M. de Falco, F. de Giovanni, C. Musella, “Groups with normality conditions for subgroups of infinite rank”, Publ. Mat., 58:2 (2014), 331–340 | DOI | MR | Zbl

[31] M. de Falco, F. de Giovanni, C. Musella, “Groups in which every normal subgroup of infinite rank has finite index”, Southeast Asian Bull. Math., 39:2 (2015), 195–201 | MR | Zbl

[32] M. de Falco, F. de Giovanni, C. Musella, “A note on groups of infinite rank with modular subgroup lattice”, Monatsh. Math., 176:1 (2015), 81–86 | DOI | MR | Zbl

[33] M. de Falco, F. de Giovanni, C. Musella, Y. P. Sysak, “Groups of infinite rank in which normality is a transitive relation”, Glasg. Math. J., 56:2 (2014), 387–393 | DOI | MR | Zbl

[34] M. de Falco, F. de Giovanni, C. Musella, Y. P. Sysak, “On metahamiltonian groups of infinite rank”, J. Algebra, 407 (2014), 135–148 | DOI | MR | Zbl

[35] M. de Falco, F. de Giovanni, C. Musella, N. Trabelsi, “Groups whose proper subgroups of infinite rank have finite conjugacy classes”, Bull. Austral. Math. Soc., 89:1 (2014), 41–48 | DOI | MR | Zbl

[36] M. de Falco, F. de Giovanni, C. Musella, N. Trabelsi, “Groups with restrictions on subgroups of infinite rank”, Rev. Mat. Iberoamericana, 30:2 (2014), 537–550 | DOI | MR | Zbl

[37] M. de Falco, F. de Giovanni, C. Musella, N. Trabelsi, “Groups of infinite rank with finite conjugacy classes of subnormal subgroups”, J. Algebra, 431 (2015), 24–37 | DOI | MR | Zbl

[38] S. Franciosi, F. de Giovanni, M. L. Newell, “Groups with polycyclic non-normal subgroups”, Algebra Colloq., 7:1 (2000), 33–42 | DOI | MR | Zbl

[39] H. Heineken, I. J. Mohamed, “A group with trivial centre satisfying the normalizer condition”, J. Algebra, 10:3 (1968), 368–376 | DOI | MR | Zbl

[40] K. Iwasawa, “On the structure of infinite M-groups”, Japan. J. Math., 18 (1943), 709–728 | MR | Zbl

[41] M. I. Kargapolov, “Some problems in the theory of nilpotent and solvable groups”, Dokl. Akad. Nauk SSSR, 127 (1959), 1164–1166 | MR | Zbl

[42] M. I. Kargapolov, “On soluble groups of finite rank”, Algebra Logika, 1:5 (1962), 37–44 | MR | Zbl

[43] O. H. Kegel, B. A. F. Wehrfritz, Locally Finite Groups, North-Holland Mathematical Library, 3, North-Holland, Amsterdam–London, 1973 | MR | Zbl

[44] L. A. Kurdachenko, H. Smith, “Groups in which all subgroups of infinite rank are subnormal”, Glasg. Math. J., 46:1 (2004), 83–89 | DOI | MR | Zbl

[45] L. A. Kurdachenko, P. Soules, “Groups with all non-subnormal subgroups of finite rank”, Groups St. Andrews 2001 in Oxford, v. II, London Math. Soc. Lecture Note, 305, 2003, 366–376 | MR | Zbl

[46] A. Lubotzky, A. Mann, “Residually finite groups of finite rank”, Math. Proc. Cambridge Philos. Soc., 106:3 (1989), 385–388 | DOI | MR | Zbl

[47] A. I. Maltsev, “On groups of finite rank”, Mat. Sb., 22 (1948), 351–352 | MR | Zbl

[48] A. I. Maltsev, “On certain classes of infinite soluble groups”, Mat. Sb., 28 (1951), 367–388

[49] Yu. I. Merzljakov, “Locally solvable groups of finite rank”, Algebra Logika, 3:2 (1964), 5–16 | MR | Zbl

[50] Yu. I. Merzlyakov, “On locally soluble groups of finite rank II”, Algebra Logika, 8:6 (1969), 686–690 | DOI | MR

[51] W. Möhres, “Auflösbarkeit von Gruppen, deren Untergruppen alle subnormal sind”, Arch. Math., 54:3 (1990), 232–235 | DOI | MR | Zbl

[52] F. Napolitani, E. Pegoraro, “On groups with nilpotent-by-Chernikov proper subgroups”, Arch. Math., 69:2 (1997), 89–94 | DOI | MR | Zbl

[53] A. Yu. Olshanskii, “Groups of bounded exponent with subgroups of prime order”, Algebra Logic, 21:5 (1982), 369–418 | DOI

[54] A. Yu. Olshanskii, Geometry of Defining Relations in Groups, Mathematics and its Applications, 70, Kluwer Academic Publishers, Dordrecht, 1991 | DOI

[55] O. Ore, “On the application of structure theory to groups”, Bull. Amer. Math. Soc., 44:12 (1938), 801–806 | DOI | MR

[56] J. Otal, J. M. Peňa, “Locally graded groups with certain minimal conditions for subgroups II”, Publ. Mat., 32:2 (1988), 151–157 | DOI | MR | Zbl

[57] J. Otal, J. M. Peňa, “Groups in which every proper subgroup is Cernikov-by-nilpotent or nilpotent-by-Cernikov”, Arch. Math., 51:3 (1988), 193–197 | DOI | MR | Zbl

[58] D. J. S. Robinson, Finiteness Conditions and Generalized Soluble Groups, v. 1 and 2, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag, Berlin–Heidelberg–New York, 1972 | MR

[59] D. J. S. Robinson, “A new treatment of soluble groups with a finiteness condition on their abelian subgroups”, Bull. London Math. Soc., 8:2 (1976), 113–129 | DOI | MR | Zbl

[60] J. E. Roseblade, “On groups in which every subgroup is subnormal”, J. Algebra, 2:4 (1965), 402–412 | DOI | MR | Zbl

[61] O. Yu. Schmidt, “Groups all of whose subgroups are nilpotent”, Mat. Sb., 31 (1924), 366–372

[62] R. Schmidt, Subgroup Lattices of Groups, de Gruyter Expositions in Mathematics, 14, Walter de Gruyter Co., Berlin, 1994 | MR | Zbl

[63] V. P. Shunkov, “On locally finite groups of finite rank”, Algebra Logic, 10:2 (1971), 127–142 | DOI | MR

[64] S. E. Stonehewer, “Permutable subgroups of infinite groups”, Math. Z., 125:1 (1972), 1–16 | DOI | MR | Zbl

[65] D. I. Zaitsev, “Stably solvable groups”, Izv. Akad. Nauk SSSR, 33 (1969), 765–780 | MR | Zbl

[66] D. I. Zaitsev, “Solvable groups of finite rank”, Groups with Restricted Subgroups, “Naukova Dumka”, Kiev, 1971, 115–130

[67] D. I. Zaitsev, “Groups with complemented normal subgroups”, Algebra Logic, 14:1 (1975), 1–6 | DOI

[68] D. I. Zaitsev, “Groups of operators of finite rank and their application”, Sixth All-Union Symposium on Group Theory ((Cherkasy, 1978)), “Naukova Dumka”, Kiev, 1980, 22–37, 219 pp.

[69] D. I. Zaitsev, “Products of abelian groups”, Algebra Logic, 19:2 (1980), 94–106 | DOI