Revista Matemática Iberoamericana

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Volume 24, Issue 2, 2008, pp. 433–461
DOI: 10.4171/RMI/542

Published online: 2008-08-31

Soluble products of connected subgroups

M. Pilar Gállego[1], Peter Hauck[2] and M. Dolores Pérez-Ramos[3]

(1) Universidad de Zaragoza, Spain
(2) Universität Tübingen, Germany
(3) Universitat de València, Burjassot (Valencia), Spain

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

Keywords: Finite groups, soluble groups, 2-generated subgroups, product of subgroups, metanilpotent groups, Fitting series

Gállego M. Pilar, Hauck Peter, Pérez-Ramos M. Dolores: Soluble products of connected subgroups. Rev. Mat. Iberoamericana 24 (2008), 433-461. doi: 10.4171/RMI/542