Open Access
July, 2008 Soluble products of connected subgroups
M. Pilar Gállego , Peter Hauck , M. Dolores Pérez-Ramos
Rev. Mat. Iberoamericana 24(2): 433-461 (July, 2008).


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 $h\in G$.


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M. Pilar Gállego . Peter Hauck . M. Dolores Pérez-Ramos . "Soluble products of connected subgroups." Rev. Mat. Iberoamericana 24 (2) 433 - 461, July, 2008.


Published: July, 2008
First available in Project Euclid: 11 August 2008

zbMATH: 1182.20025
MathSciNet: MR2459199

Primary: 20D10 , 20D40

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

Rights: Copyright © 2008 Departamento de Matemáticas, Universidad Autónoma de Madrid

Vol.24 • No. 2 • July, 2008
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