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Characterisation of finitely generated soluble finite-by-nilpotent groups. (English) Zbl 1119.20039
For a class of groups \(\mathfrak X\), let \((\mathfrak X,\infty)\) denote the class of groups in which every infinite subset contains two distinct elements generating a subgroup in \(\mathfrak X\). B. H. Neumann [J. Aust. Math. Soc., Ser. A 21, 467-472 (1976; Zbl 0333.05110)] proved, for the class \(\mathfrak A\) of Abelian groups, that the groups in \((\mathfrak A,\infty)\) are exactly the center-by-finite groups. J. C. Lennox and J. Wiegold [ibid. 31, 459-463 (1981; Zbl 0492.20019)] showed, for the class \(\mathfrak N\) of nilpotent groups, that a finitely generated soluble group is in \((\mathfrak N,\infty)\) iff it is finite-by-nilpotent.
Here are the main results of the paper under review. For \(k>0\), let \(\Omega_k\) be the class of groups \(H\) such that \(\gamma_{k-1}(H)>\gamma_k(H)=\gamma_{k+1}(H)\) (where \(\gamma_i(H)\) is the \(i\)-th member of the lower central series of \(H\)), and \(\Omega=\bigcup_k\Omega_k\). Let \(G\) be a finitely generated soluble group. Then (1) \(G\in(\Omega,\infty)\) iff \(G\) is finite-by-nilpotent, and (2) \(G\in(\Omega_k,\infty)\) iff \(G\) is finite-by-\(\mathfrak N_{k,2}\), where \(\mathfrak N_{k,2}\) is the class of groups in which every two-generated subgroup is \(k\)-step nilpotent.

MSC:
20F16 Solvable groups, supersolvable groups
20F05 Generators, relations, and presentations of groups
20F14 Derived series, central series, and generalizations for groups
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References:
[1] A. ABDOLLAHI, A characterization of infinite 3-abelian groups, Arch. Math. (Basel), 73 (1999), pp. 104-108. Zbl0936.20038 MR1703676 · Zbl 0936.20038
[2] A. ABDOLLAHI - B. TAERI, Some conditions on infinite subsets of infinite groups, Bull. Malaysian Math. Soc. (Second Series), 22 (1999), pp. 87-93. Zbl1006.20034 MR1747663 · Zbl 1006.20034
[3] A. ABDOLLAHI - B. TAERI, A condition on finitely generated soluble groups, Comm. Algebra, 27 (1999), pp. 5633-5638. Zbl0942.20014 MR1713058 · Zbl 0942.20014
[4] A. ABDOLLAHI - B. TAERI, A condition on a certain variety of groups, Rend. Sem. Mat. Univ. Padova, 104 (2000), pp. 129-134. Zbl1013.20021 MR1809354 · Zbl 1013.20021
[5] A. ABDOLLAHI - N. TRABELSSI, Quelques extentions d’un problème de Paul Erdos sur les groupes, Bull. Belg. Math. Soc. Zbl1041.20022 · Zbl 1041.20022
[6] R. BAER, Representations of groups as quotient groups II, Trans. Amer. Soc, 58 (1945), pp. 348-389. Zbl0061.02703 MR15107 · Zbl 0061.02703
[7] A. BOUKAROURA, Two conditions for infinite groups to satisfy certain laws, Algebra Colloq., 10:1 (2003), pp. 75-80. Zbl1031.20018 MR1961508 · Zbl 1031.20018
[8] C. DELIZIA - H. SMITH - A. H. RHAMTULLA, Locally graded groups with a nilpotency condition on infinite subsets, J. Austral. Math. Soc. A, 69 (2000), pp. 415-420. Zbl0982.20019 MR1793472 · Zbl 0982.20019
[9] G. ENDIMIONI, Groupes finis satisfaisant la condition ], n), C. R. Acad. Sci. Paris. Série I, 319 (1994), pp. 1245-1247. Zbl0822.20023 MR1310664 · Zbl 0822.20023
[10] K. W. GRUENBERG, Residual properties of infinite soluble groups, Proc. London Math. Soc., 7 (1957), pp. 29-62. Zbl0077.02901 MR87652 · Zbl 0077.02901
[11] P. HALL, Finite by nilpotent groups, Proc. Combridge Phlos. Soc., 52 (1956), pp. 611-616. Zbl0072.25801 MR80095 · Zbl 0072.25801
[12] O. H. KEGEL - B. A. F. WEHRFRITS, Locally finite groups, North-Holland, Amsterdam 1973. Zbl0259.20001 · Zbl 0259.20001
[13] P. S. KIM - A. H. RHEMTULLA - H. SMITH, A characterization of infinite metabelian groups, Houston J. Math., 17 (1991), pp. 129-137. Zbl0744.20033 · Zbl 0744.20033
[14] J. C. LENNOX - J. WIEGOLD, Extension of a problem of Paul Erdos on groups, J. Austral. Math. Soc., 31 (1981), pp. 459-463. Zbl0492.20019 MR638274 · Zbl 0492.20019
[15] P. LONGOBARDI - M. MAJ, A finiteness condition concerning commutators in groups, Houston J. Math., 19 (4) (1993), pp. 505-512. Zbl0813.20026 MR1251605 · Zbl 0813.20026
[16] P. LONGOBARDI - M. MAJ - A. H. RHEMTULLA, Infinite groups in a given variety and Ramsey’s theorem, Comm. Algebra, 20 (1992), pp. 127-139. Zbl0751.20020 MR1145329 · Zbl 0751.20020
[17] B. H. NEUMANN, A problem of Paul Erdos on groups, J. Austral. Math. Soc. Ser. A, 21 (1976), pp. 467-472. Zbl0333.05110 MR419283 · Zbl 0333.05110
[18] D. J. S. ROBINSON, A course in the theory of groups, Springer-Verlag, New York, 1982. Zbl0483.20001 MR648604 · Zbl 0483.20001
[19] D. SEGAL, A residual property of finitely generated abelian by nilpotent groups, J. Algebra, 32 (1974), pp. 389-399. Zbl0293.20029 MR419612 · Zbl 0293.20029
[20] L. S. SPIEZIA, A characterization of the third Engel groups, Arch. Math. (Basel), 64 (1995), pp. 369-373. Zbl0823.20038 MR1323912 · Zbl 0823.20038
[21] B. TAERI, A Question of Paul Erdos and nilpotent-by-finite groups, Bull. Austral. Math. Soc., 64 (2001), pp. 245-254. Zbl0995.20020 MR1860061 · Zbl 0995.20020
[22] N. TRABELSSI, Characterisation of nilpotent-by-finite groups, Bull. Austral. Math. Soc., 61 (2000), pp. 33-38. Zbl0959.20034 MR1819313 · Zbl 0959.20034
[23] B. A. F. WEHRFRITS, Infinite linear groups (Springer, 1973). Zbl0261.20038 · Zbl 0261.20038
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