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PFMT, 2017, Issue 2(31), Pages 46–49 (Mi pfmt501)  

MATHEMATICS

A criterion for a finite group to belong a saturated formation

I. M. Dergacheva, I. P. Shabalina, E. A. Zadorozhnyuk

Belarusian State University of Transport

Abstract: We prove the following result: Let $\mathcal{F}$ be a hereditary saturated formation of $p$-soluble groups containing all $p$-supersoluble groups such that $\mathcal{F}=\mathcal{G}_p\mathcal{F}$. Let $G=AT$ where $A$ is a Hall $\pi$-subgroup of $G$, $p\notin\pi$ and $T$ is a $p$-supersoluble subgroup of $G$. Suppose that for a Sylow $p$-subgroup $P$ of $T$ we have $|P|>p$. If $A$ permutes with a Hall $p'$-subgroup of $T$ and with all maximal subgroups $V$ of $P$ such that $G^{\mathcal{F}}\cap P\not\leqslant V$, then $G\in\mathcal{F}$.

Keywords: finite group, saturated formation, $p$-soluble group, $p$-supersoluble group, Hall subgroup.

Full text: PDF file (330 kB)
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UDC: 512.542
Received: 29.04.2017
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Citation: I. M. Dergacheva, I. P. Shabalina, E. A. Zadorozhnyuk, “A criterion for a finite group to belong a saturated formation”, PFMT, 2017, no. 2(31), 46–49

Citation in format AMSBIB
\Bibitem{DerShaZad17}
\by I.~M.~Dergacheva, I.~P.~Shabalina, E.~A.~Zadorozhnyuk
\paper A criterion for a finite group to belong a saturated formation
\jour PFMT
\yr 2017
\issue 2(31)
\pages 46--49
\mathnet{http://mi.mathnet.ru/pfmt501}


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