Countably recognizable classes of groups with restricted conjugacy classes

Document Type : Ischia Group Theory 2016


1 Dipartimento di Matematica e Applicazioni - University of Napoli "Federico II"

2 Universita di Napoli Federico II,


A group class ${ X}$ is said to be countably recognizable if a group belongs to ${X}$ whenever all its countable subgroups lie in ${X}$‎. ‎It is proved here that most of the relevant classes of groups defined by restrictions on the conjugacy classes are countably recognizable‎.


Main Subjects

[1] R. Baer, Abzählbar erkennbare gruppentheoretische Eigenschaften, Math. Z., 79 (1962) 344–363.
[2] J. T. Buckley, J. C. Lennox, B. H. Neumann, H. Smith and J. Wiegold, Groups with all subgroups normal-by-finite, J. Austral. Math. Soc. Ser. A, 59 (1995) 384–398.
[3] F. Catino and F. de Giovanni, Some Topics in the Theory of Groups with Finite Conjugacy Classes, 1, Aracne, Roma, 2015.
[4] M. R. Dixon, M. J. Evans and H. Smith, Some countably recognizable classes of groups, J. Group Theory, 10 (2007) 641–653.
[5] S. Franciosi, F. de Giovanni and M. J. Tomkinson, Groups with polycyclic-by-finite conjugacy classes, Boll. Un. Mat. Ital. B (7), 4 (1990) 35–55.
[6] L. Fuchs, Infinite Abelian Groups, 1, Academic Press, New York, 1970.
[7] F. de Giovanni, M. Martusciello and C. Rainone, Locally finite groups whose subgroups have finite normal oscillation, Bull. Aust. Math. Soc., 89 (2014) 479–487.
[8] F. de Giovanni, A. Russo and G. Vincenzi, Groups with restricted conjugacy classes, Serdica Math. J., 28 (2002) 241–254.
[9] F. de Giovanni and M. Trombetti, Countable recognizability and nilpotency properties of groups, Rend. Circa. Mat. Palermo, to appear.
[10] G. Higman, Almost free groups, Proc. London Math. Soc. (3), 1 (1951) 284–290.
[11] M. I. Kargapolov, Some problems in the theory of nilpotent and solvable groups, Dokl. Akad. Nauk SSSR, 127 (1959) 1164–1166.
[12] D. H. McLain, Remarks on the upper central series of a group, Proc. Glasgow Math. Assoc., 3 (1956) 38–44.
[13] B. H. Neumann, Groups covered by permutable subsets, J. London Math. Soc., 29 (1954) 236–248.
[14] B. H. Neumann, Groups with finite classes of conjugate subgroups, Math. Z., 63 (1955) 76–96.
[15] B. H. Neumann, Group properties of countable character, Selected questions of algebra and logic (collection dedicated to the memory of A. I. Mal’cev), Nauka, Novosibirsk, 1973 197–204.
[16] R. E. Phillips, f-systems in infinite groups, Arch. Math. (Basel), 20 (1969) 345–355.
[17] R. E. Phillips, Countably recognizable classes of groups, Rocky Mountain J. Math., 1 (1971) 489–497.
[18] Y. D. Polovicki˘ı, Groups with extremal classes of conjugate elements, Sibirsk. Mat. Z., 5 (1964) 891–895.
[19] D. J. S. Robinson, Finiteness Conditions and Generalized Soluble Groups, Springer, Berlin, 1972.
[20] D. J. S. Robinson, On groups with extreme centralizers and normalizers, Adv. Group Theory Appl., 1 (2016) 97–112.
[21] A. Russo and G. Vincenzi, Groups with many generalized FC-subgroups, Algebra Discrete Math., (2009) 158–166.
[22] H. Smith, More countably recognizable classes of groups, J. Pure Appl. Algebra, 213 (2009) 1320–1324.
[23] H. Smith and J. Wiegold, Locally graded groups with all subgroups normal-by-finite, J. Austral. Math. Soc. Ser. A, 60 (1996) 222–227.
[24] M. J. Tomkinson, FC-groups, 96, Pitman, Boston, 1984.