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 Sib. Èlektron. Mat. Izv., 2009, Volume 6, Pages 366–380 (Mi semr72)

This article is cited in 6 scientific papers (total in 6 papers)

Research papers

Around a conjecture of P. Hall

D. O. Revinab

a Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences
b Novosibirsk State University

Abstract: In the paper, we discuss perspectives of future investigations of the Hall $\pi$-properties $E_\pi$, $C_\pi$ and $D_\pi$ in finite groups. A series of open problems is stated, both comparatirely new and well-known ones. It is proven that there are infinitely many infinite sets $\pi$ of primes with $E_\pi\Rightarrow D_\pi$. Precisely if $\pi$ consists of the primes $p>x$, for every real $x\ge7$ then $E_\pi\Rightarrow D_\pi$. This result continues the investigations initiated by well-known Hall's conjecture of 1956 that $E_\pi\Rightarrow D_\pi$ for every set $\pi$ of odd primes. This conjecture was disproved by F. Gross, who showed in 1984 that, for every finite set $\pi$ of odd primes with $|\pi|\ge2$, there exists a finite group $G$ such that $G\in E_\pi$ and $G\notin D_\pi$.

Keywords: prime number, $\pi$-subgroup, $\pi$-Hall subgroup, properties $E_\pi$, $C_\pi$ and $D_\pi$.

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Document Type: Article
UDC: 512.542
MSC: 20D20
Received September 3, 2009, published November 7, 2009

Citation: D. O. Revin, “Around a conjecture of P. Hall”, Sib. Èlektron. Mat. Izv., 6 (2009), 366–380

Citation in format AMSBIB
\Bibitem{Rev09} \by D.~O.~Revin \paper Around a~conjecture of P.~Hall \jour Sib. \Elektron. Mat. Izv. \yr 2009 \vol 6 \pages 366--380 \mathnet{http://mi.mathnet.ru/semr72} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2586695} `

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This publication is cited in the following articles:
1. E. P. Vdovin, D. O. Revin, “Theorems of Sylow type”, Russian Math. Surveys, 66:5 (2011), 829–870
2. D. O. Revin, “On a relation between the Sylow and Baer–Suzuki theorems”, Siberian Math. J., 52:5 (2011), 904–913
3. E. P. Vdovin, D. O. Revin, “Pronormality of Hall subgroups in finite simple groups”, Siberian Math. J., 53:3 (2012), 419–430
4. E. P. Vdovin, D. O. Revin, “On the pronormality of Hall subgroups”, Siberian Math. J., 54:1 (2013), 22–28
5. W. Guo, D. O. Revin, “On the class of groups with pronormal Hall $\pi$-subgroups”, Siberian Math. J., 55:3 (2014), 415–427
6. W. Guo, D. O. Revin, “Maximal and submaximal $\mathfrak X$-subgroups”, Algebra and Logic, 57:1 (2018), 9–28
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