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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: crestani@math.unipd.it, lucchini@math.unipd.it
ANDREA LUCCHINI
Affiliation:
Dipartimento di Matematica, Via Trieste 63, 35121 Padova, Italy e-mail: crestani@math.unipd.it, lucchini@math.unipd.it
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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

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