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Groups with a nilpotent triple factorisation

Published online by Cambridge University Press:  17 April 2009

Bernhard Amberg ,
Silvana Franciosi  and
Francesco de Giovanni
Bernhard Amberg
Affiliation:
Fachbereich Mathematik, Universität Mainz, Saarstraße 21, D-6500 Mainz, West Germany.
Silvana Franciosi
Affiliation:
Dipartimento di Matematica, Università di Napoli, via Mezzocannone 8, I - 80134 Napoli, Italy.
Francesco de Giovanni
Affiliation:
Dipartimento di Matematica, Università di Napoli, via Mezzocannone 8, I - 80134 Napoli, Italy.
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Abstract

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In the investigation of factorised groups one often encounters groups G = AB = AK = BK which have a triple factorisation as a product of two subgroups A and B and a normal subgroup K of G. It is of particular interest to know whether G satisfies some nilpotency requirement whenever the three subgroups A, B and K satisfy this same nilpotency requirement. A positive answer to this problem for the classes of nilpotent, hypercentral and locally nilpotent groups is given under the hypothesis that K is a minimax group or G has finite abelian section rank. The results become false if K has only finite Prüfer rank. Some applications of the main theorems are pointed out.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 37 , Issue 1 , February 1988 , pp. 69 - 79
Copyright
Copyright © Australian Mathematical Society 1988

References

[1]Amberg, B., ‘Artinian and noetherian factorized groups’, Rend. Sem. Mat. Univ. Padova 55 (1976), 105–122.Google Scholar
[2]Amberg, B., ‘Lokal endlich-auflösbare Produkte von zwei hyperzentralen Gruppen’, Arch. Math. (Basel) 35 (1980), 228–238.CrossRefGoogle Scholar
[3]Amberg, B., ‘Uber den Satz von Kegel und Wielandt’, Arch. Math. (Basel) 40 (1983), 289–296.CrossRefGoogle Scholar
[4]Amberg, B., ‘Products of two abelian subgroups’, Rocky Mountain J. Math. 14 (1984), 541–547.CrossRefGoogle Scholar
[5]Amberg, B., ‘On groups which are the product of abelian subgroups’, Glasgow Math. J. 28 (1985), 151–156.CrossRefGoogle Scholar
[6]Amberg, B., ‘Produkte von Gruppen mit endlichem torsionfreiem Rang’, Arch. Math. (Basel) 45 (1985), 398–406.CrossRefGoogle Scholar
[7]Amberg, B., ‘Products of groups of finite rank’, Arch. Math. (Basel) (to appear).Google Scholar
[8]Amberg, B. and Robinson, D.J.S., ‘Soluble groups which are products of nilpotent minimax groups’, Arch. Math. (Basel) 42 (1984), 385–390.CrossRefGoogle Scholar
[9]Černikov, N.S., ‘Products of groups of finite free rank’, in Groups and systems of their subgroups, pp. 42–56 (Inst. Math., Kiev, 1983).Google Scholar
[10]Halbritter, N., ‘Zur Kohomologie von lokal nilpotenten Gruppen’ (Diplomarbeit, Mainz).Google Scholar
[11]Itô, N., ‘Über das Produkt von zwei abelschen Gruppen’, Math. Z. 62 (1955), 400–401.CrossRefGoogle Scholar
[12]Kegel, O.H., ‘Produkte nilpotenten Gruppen’, Arch. Math. (Basel) 12 (1961), 90–93.CrossRefGoogle Scholar
[13]Pennington, E., ‘On products of finite nilpotent groups’, Math. Z. 134 (1973), 31–83.CrossRefGoogle Scholar
[14]Robinson, D.J.S., ‘A property of the lower central series of a group’, Math. Z. 107 (1968), 225–231.CrossRefGoogle Scholar
[15]Robinson, D.J.S., Finiteness conditions and generalized soluble groups (Springer, Berlin, 1972).CrossRefGoogle Scholar
[16]Robinson, D.J.S., ‘Soluble products of nilpotent groups’, J. Algebra 98 (1986), 183–196.CrossRefGoogle Scholar
[17]Robinson, D.J.S., ‘Cohomology of locally nilpotent groups’, J. Pure Appl. Algebra (to appear).Google Scholar
[18]Sysak, Y.P., ‘Products of infinite groups’, (preprint 82.53), Akad. Nauk Ukr. SSR (1982).Google Scholar
[19]Wielandt, H., ‘Über Produkte von nilpotenten Gruppen’, Illinois J. Math. 2 (1958), 611–618.CrossRefGoogle Scholar
[20]Zaicev, D.I., ‘Products of abelian groups’, Algebra and Logic 19 (1980), 94–106.Google Scholar
[21]Zaicev, D.I., ‘Nilpotent approximations of metabelian groups’, Algebra and Logic 20 (1981), 413–423.Google Scholar