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LARGE P-GROUPS WITHOUT PROPER SUBGROUPS WITH THE SAME DERIVED LENGTH

Part of: Representation theory of groups

Published online by Cambridge University Press:  23 July 2015

ELEONORA CRESTANI  and
ANDREA LUCCHINI
ELEONORA CRESTANI
Affiliation:
Dipartimento di Matematica, Via Trieste 63, 35121 Padova, Italy e-mail: [email protected], [email protected]
ANDREA LUCCHINI
Affiliation:
Dipartimento di Matematica, Via Trieste 63, 35121 Padova, Italy e-mail: [email protected], [email protected]
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Abstract

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We construct a subgroup Hd of the iterated wreath product Gd of d copies of the cyclic group of order p with the property that the derived length and the smallest cardinality of a generating set of Hd are equal to d while no proper subgroup of Hd has derived length equal to d. It turns out that the two groups Hd and Gd are the extreme cases of a more general construction that produces a chain Hd=K1<···< Kp−1=Gd of subgroups sharing a common recursive structure. For i ∈ {1,. . .,p−1}, the subgroup Ki has nilpotency class (i+1)d−1.

MSC classification

Primary: 20D15: Nilpotent groups, $p$-groups
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 58 , Issue 2 , May 2016 , pp. 445 - 459
Copyright
Copyright © Glasgow Mathematical Journal Trust 2015 

References

REFERENCES

1.Evans-Riley, S., Newman, M. F. and Schneider, C., On the soluble length of groups with prime-power order, Bull. Austral. Math. Soc. 59 (2) (1999), 343–346.Google Scholar
2.Glasby, S. P., The shape of solvable groups with odd order, Groups St. Andrews 2005. Vol. 2, London Math. Soc. Lecture Note Ser., vol. 340, (Cambridge University Press, Cambridge, 2007), 432–437.Google Scholar
3.Huppert, B., Endliche Gruppen. I, Die Grundlehren der Mathematischen Wissenschaften, Band, vol. 134, (Springer-Verlag, Berlin, 1967).CrossRefGoogle Scholar
4.Mann, A., The derived length of p-groups, J. Algebra 224 (2) (2000), 263–267.Google Scholar
5.Schneider, C., The derived series of a finite p-group, J. Algebra 307 (1) (2007), 136–152.CrossRefGoogle Scholar