JournalsrmiVol. 24, No. 2pp. 433–461

Soluble products of connected subgroups

  • M. Pilar Gállego

    Universidad de Zaragoza, Spain
  • Peter Hauck

    Universität Tübingen, Germany
  • M. Dolores Pérez-Ramos

    Universitat de València, Burjassot (Valencia), Spain
Soluble products of connected subgroups cover
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Abstract

The main result in the paper states the following: For a finite group G=ABG=AB, which is the product of the soluble subgroups AA and BB, if a,b\langle a,b \rangle is a metanilpotent group for all aAa\in A and bBb\in B, then the factor groups a,bF(G)/F(G)\langle a,b \rangle F(G)/F(G) are nilpotent, F(G)F(G) denoting the Fitting subgroup of GG. A particular generalization of this result and some consequences are also obtained. For instance, such a group GG is proved to be soluble of nilpotent length at most l+1l+1, assuming that the factors AA and BB have nilpotent length at most ll. Also for any finite soluble group GG and k1k\geq 1, an element gGg\in G is contained in the preimage of the hypercenter of G/Fk1(G)G/F_{k-1}(G), where Fk1(G)F_{k-1}(G) denotes the (k1k-1)th term of the Fitting series of GG, if and only if the subgroups g,h\langle g,h\rangle have nilpotent length at most kk for all hGh\in G.

Cite this article

M. Pilar Gállego, Peter Hauck, M. Dolores Pérez-Ramos, Soluble products of connected subgroups. Rev. Mat. Iberoam. 24 (2008), no. 2, pp. 433–461

DOI 10.4171/RMI/542