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Fibered p-groups. (English) Zbl 0618.20016
Suppose G is a p-group which is not necessarily finite but has non- trivial centre Z. The paper is concerned with the case when G has a partition, that is, a set of at least two subgroups for which any non- identity element of G lies in precisely one of them. The case of exponent p is not of much interest, since the set of all subgroups of order p is such a partition, but it is easy to see that if the exponent of G is greater than p, the subgroup \(H_ p\) generated by all elements of order greater than p is contained in a member of the partition. Thus \(H_ p<G\), and conversely, groups for which \(H_ p<G\) have an obvious partition. Consideration of such groups leads to the definition of a group of type (p,\(\nu)\) as \(<a,b>\), when \(| <a>| =p^{\nu}\), ab\(\neq ba\), \(b\not\in <a>^ G\), and \((b^ ix)=1\) for \((i,p)=1\) and \(x\in <a>^ G\). Such groups are described in terms of a class of commutative local rings. In the final section, some examples are constructed by means of couplings. The readability of the paper is marred by its unorthodox notation.
Reviewer: N.Blackburn

20D15 Finite nilpotent groups, \(p\)-groups
20D60 Arithmetic and combinatorial problems involving abstract finite groups
20D30 Series and lattices of subgroups
16Y30 Near-rings
Full Text: DOI
[1] A. Herzer, Endliche nicht kommutative Gruppen mit Partition \(\pi\) und fixpunkt-freiem \(\pi\)-Automorphismus. Arch. d. Math.34 (1980), 385–392. · Zbl 0431.51001 · doi:10.1007/BF01224975
[2] H. Karzel, Unendliche Dicksonsche Fastkörper. Arch. Math.16 (1965), 247–2561. · Zbl 0131.01701 · doi:10.1007/BF01220030
[3] H. Karzel, Kinematische Algebren und ihre geometrischen Ableitungen. Abh. Math. Sem. Univ. Hamburg41 (1974), 158–171. · Zbl 0296.50019 · doi:10.1007/BF02993509
[4] H. Karzel, Affine incidence groups. Rendiconti del Seminario Matematico di Brescia, Vol.7 (1984), 409–425. Atti del Convegno ”Geometria combinatoria e di incidenza: fondamenti e applicazioni”, La Mendola, 4–11 luglio 1982.
[5] H. Karzel andJ. C. Maxson, Kinematic spaces with dilatations. J. of Geometry22 (1984), 196–201. · Zbl 0537.51022 · doi:10.1007/BF01222846
[6] H. Karzel andJ. C. Maxson, Fibered groups with non-trivial centers. Res. d. Mathematik7 (1984), 192–208. · Zbl 0553.20011
[7] M. Marchi andC. Perelli Cippo, Su una perticolare classe di S-spazi. Rend. Sem. Mat. Brescia4 (1979), 3–42.
[8] K. Sörensen, Inzidenzgruppen der Ordnung 3. Math. Zeitschr.154 (1977), 239 bis 241 · Zbl 0365.50005
[9] M. Bilotti andA. Scarselli, Sulle strutture di translazione di dilatazioni proprie. Atti Acc. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Nat. (8)67 (1979), 75–80. · Zbl 0471.51002
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