Hostname: page-component-848d4c4894-2pzkn Total loading time: 0 Render date: 2024-05-13T22:45:52.678Z Has data issue: false hasContentIssue false
  • English
  • Français

Groups with few non-nilpotent subgroups

Published online by Cambridge University Press:  18 May 2009

Howard Smith
Howard Smith
Affiliation:
Department of Mathematics, Bucknell University, Lewisburg PA 17837, U.S.A.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 39 , Issue 2 , May 1997 , pp. 141 - 151
Copyright
Copyright © Glasgow Mathematical Journal Trust 1997

References

REFERENCES

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.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.Hall, P., Nilpotent groups, Collected works of Philip Hall (Clarendon Press, Oxford, 1988), 415–462.Google Scholar
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.Heineken, H. and Mohamed, I. J., A group with trivial centre satisfying the normalizer condition, J. Algebra 10 (1968), 368–376.CrossRefGoogle Scholar
6.Kargopolov, M. I. and Merzlyakov, Yu. I., Fundamentals of the theory of groups (Springer, Berlin-Heidelberg-New York, 1979).Google Scholar
7.Kegel, O. H. and Wehrfritz, B. A. F., Locally finite groups (North Holland, 1973).Google Scholar
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.Menegazzo, F., Groups of Heineken-Mohamed, J. Algebra 171 (1995), 807–825.CrossRefGoogle Scholar
10.Möhres, W., Torsionsfreie Gruppen, deren Untergruppen alle subnormal sind, Math. Ann. 84 (1989), 245–249.Google Scholar
11.Möhres, W., Auflösbarkeit von Gruppen, deren Untergruppen alle subnormal sind, Archiv der Math. 54 (1990), 232–235.Google Scholar
12.Newman, M. F. and Wiegold, J., Groups with many nilpotent subgroups, Archiv der Math. 15 (1964), 241–250.Google Scholar
13.Yu, A.. Ol'shanskii, Geometry of defining relations in groups (Nauka, Moscow, 1989).Google Scholar
14.Rédei, L., Die endlichen einstufig nichtnilpotenten Gruppen, Publ. Math. Debrecen 4 (1956), 130–138.Google Scholar
15.Robinson, D. J. S., Finiteness conditions and generalized soluble groups (2 vols.) (Springer, Berlin-Heidelberg-New York, 1972).Google Scholar
16.Robinson, D. J. S., A course in the theory of groups (Springer, Berlin-Heidelberg-New York, 1982).CrossRefGoogle Scholar
17.Robinson, D. J. S., Soluble products of nilpotent groups, J. Algebra 98 (1986), 183–196.Google Scholar
18.Yu, O.. Schmidt, Groups all of whose subgroups are nilpotent, Mat. Sbornik 31 (1924), 366–372. (Russian).Google Scholar
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.Wilson, J. S., Some properties of groups inherited by normal subgroups of finite index, Math. Z. 114 (1970), 19–21.Google Scholar
21.Zaicev, D. I., Theory of minimax groups, Ukrainian Math. J. 23 (1971), 536–542.Google Scholar