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On the product of two $\pi$-decomposable soluble groups

L. S. Kazarin, A. Martínez-Pastor, and M. D. Pérez-Ramos

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Let the group $G=AB$ be a product of two $\pi$-decomposable subgroups $A=O_\pi(A) \times O_{\pi'}(A)$ and $B=O_\pi(B) \times O_{\pi'}(B)$ where $\pi$ is a set of primes. The authors conjecture that $O_\pi(A)O_\pi(B)=O_\pi(B)O_\pi(A)$ if $\pi$ is a set of odd primes. In this paper it is proved that the conjecture is true if $A$ and $B$ are soluble. A similar result with certain additional restrictions holds in the case $2 \in \pi$. Moreover, it is shown that the conjecture holds if $O_{\pi'}(A)$ and $O_{\pi'}(B)$ have coprime orders.

Article information

Publ. Mat., Volume 53, Number 2 (2009), 439-456.

First available in Project Euclid: 20 July 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20D20: Sylow subgroups, Sylow properties, $\pi$-groups, $\pi$-structure 20D40: Products of subgroups

Products of groups $\pi$-decomposable groups Hall subgroups


Kazarin, L. S.; Martínez-Pastor, A.; Pérez-Ramos, M. D. On the product of two $\pi$-decomposable soluble groups. Publ. Mat. 53 (2009), no. 2, 439--456.

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