The derived series of a finite p-group

Schneider, Csaba

doi:10.1016/j.jalgebra.2006.07.013

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“…The interest on p-groups without proper subgroups with the same derived length has been related with the problem of bounding the order of a finite p-group in terms of its derived length (a long history starting from Burnside's papers, see [5] for more details). Mann [4] showed that if G is a finite p-group, then G (d) = 1 implies log p |G| > 2 d + 2d − 2.…”

Section: Introductionmentioning

confidence: 99%

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

Crestani¹,

Lucchini²

2015

Glasgow Math. J.

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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.

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Section: Introductionmentioning

confidence: 99%