Geodesic
Groups with few non-nilpotent subgroups
Glasgow mathematical journal, Tome 39 (1997) no. 2, pp. 141-151

Voir la notice de l'article provenant de la source Cambridge University Press

Let G be a non-nilpotent group in which all proper subgroups are nilpotent. If G is finite then G is soluble [18], and a classification of such groups is given in [14]. The paper [12]. of Newman and Wiegold discusses infinite groups with this property. Clearly such a group is either finitely generated or locally nilpotent. Many interesting results concerning the finitely generated case are established in [12]. Since the publication of that paper there have appeared the examples due to Ol'shanskii and Rips (see [13]) of finitely generated infinite simple p-groups all of whose proper nontrivial subgroups have order p, a prime. Following [12], let us say that a group G is an AN-group if it is locally nilpotent and non-nilpotent with all proper subgroups nilpotent. A complete description is given in Section 4 of [12] of AN-groups having maximal subgroups. Every soluble AN-gvoup has locally cyclic derived factor group and is a p-group for some prime p ([12; Lemma 4.2]). The only further information provided in [12] on AN-groups without maximal subgroups is that they are countable. Four years or so after the publication of [12], there appeared the examples of Heineken and Mohamed [5]: for every prime p there exists a metabelian, non-nilpotent p-group G having all proper subgroups nilpotent and subnormal; further, G has no maximal subgroups and so G/G' is a Prüfer p-group in each case. 
Smith, Howard. Groups with few non-nilpotent subgroups. Glasgow mathematical journal, Tome 39 (1997) no. 2, pp. 141-151. doi: 10.1017/S0017089500032031
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[1] 1.Bruno, B. and Phillips, R. E., On multipliers of Heineken-Mohamed type groups, Rend. Sem. Mat. Univ. Padova 85 (1991), 133–146. Google Scholar

[2] 2.Dixon, J. D., Sautoy, M. P. F. du, Mann, A. and Segal, D., Analytic pro-p groups, London Math. Soc. Lecture Note Series 157 (1991). Google Scholar

[3] 3.Hall, P., Nilpotent groups, Collected works of Philip Hall (Clarendon Press, Oxford, 1988), 415–462. Google Scholar

[4] 4.Hall, P., Some sufficient conditions for a group to be nilpotent, Collected works of Philip Hall (Clarendon Press, Oxford, 1988), 479–495. Google Scholar

[5] 5.Heineken, H. and Mohamed, I. J., A group with trivial centre satisfying the normalizer condition, J. Algebra 10 (1968), 368–376. Google Scholar | DOI

[6] 6.Kargopolov, M. I. and Merzlyakov, Yu. I., Fundamentals of the theory of groups (Springer, Berlin-Heidelberg-New York, 1979). Google Scholar

[7] 7.Kegel, O. H. and Wehrfritz, B. A. F., Locally finite groups (North Holland, 1973). Google Scholar

[8] 8.Kurdachenko, L. A. and Goretskij, V. Eh., Groups with weak minimality and maximality conditions for subgroups which are not normal, Ukrainian Math. J. 41 (1989), 1474–1477. Google Scholar

[9] 9.Menegazzo, F., Groups of Heineken-Mohamed, J. Algebra 171 (1995), 807–825. Google Scholar | DOI

[10] 10.Möhres, W., Torsionsfreie Gruppen, deren Untergruppen alle subnormal sind, Math. Ann. 84 (1989), 245–249. Google Scholar

[11] 11.Möhres, W., Auflösbarkeit von Gruppen, deren Untergruppen alle subnormal sind, Archiv der Math. 54 (1990), 232–235. Google Scholar

[12] 12.Newman, M. F. and Wiegold, J., Groups with many nilpotent subgroups, Archiv der Math. 15 (1964), 241–250. Google Scholar

[13] 13.Yu, A.. Ol'shanskii, Geometry of defining relations in groups (Nauka, Moscow, 1989). Google Scholar

[14] 14.Rédei, L., Die endlichen einstufig nichtnilpotenten Gruppen, Publ. Math. Debrecen 4 (1956), 130–138. Google Scholar

[15] 15.Robinson, D. J. S., Finiteness conditions and generalized soluble groups (2 vols.) (Springer, Berlin-Heidelberg-New York, 1972). Google Scholar

[16] 16.Robinson, D. J. S., A course in the theory of groups (Springer, Berlin-Heidelberg-New York, 1982). Google Scholar | DOI

[17] 17.Robinson, D. J. S., Soluble products of nilpotent groups, J. Algebra 98 (1986), 183–196. Google Scholar

[18] 18.Yu, O.. Schmidt, Groups all of whose subgroups are nilpotent, Mat. Sbornik 31 (1924), 366–372. (Russian). Google Scholar

[19] 19.Smith, H., Groups with finitely many conjugacy classes of subgroups of large derived length, Bollettino U.M.I. 9 (1995), 167–175. Google Scholar

[20] 20.Wilson, J. S., Some properties of groups inherited by normal subgroups of finite index, Math. Z. 114 (1970), 19–21. Google Scholar

[21] 21.Zaicev, D. I., Theory of minimax groups, Ukrainian Math. J. 23 (1971), 536–542. Google Scholar

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