Revista Matemática Iberoamericana

Soluble products of connected subgroups

M. Pilar Gállego , Peter Hauck , and M. Dolores Pérez-Ramos

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Abstract

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 $h\in G$.

Article information

Source
Rev. Mat. Iberoamericana, Volume 24, Number 2 (2008), 433-461.

Dates
First available in Project Euclid: 11 August 2008

Permanent link to this document
https://projecteuclid.org/euclid.rmi/1218475349

Mathematical Reviews number (MathSciNet)
MR2459199

Zentralblatt MATH identifier
1182.20025

Subjects
Primary: 20D10: Solvable groups, theory of formations, Schunck classes, Fitting classes, $\pi$-length, ranks [See also 20F17] 20D40: Products of subgroups

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

Citation

Gállego, M. Pilar; Hauck, Peter; Pérez-Ramos, M. Dolores. Soluble products of connected subgroups. Rev. Mat. Iberoamericana 24 (2008), no. 2, 433--461. https://projecteuclid.org/euclid.rmi/1218475349


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