
Equivalence of the existence of nonconjugate and nonisomorphic Hall $\pi$subgroups
Guo Wen Bin^{a}, A. A. Buturlakin^{bc}, D. O. Revin^{bca} ^{a} School of Mathematical Sciences, University of Science and Technology of China
^{b} Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk
^{c} Novosibirsk State University
Abstract:
Let $\pi$ be some set of primes. A subgroup $H$ of a finite group $G$ is called a Hall $\pi$subgroup if any prime divisor of the order $H$ of the subgroup $H$ belongs to $\pi$ and the index $G:H$ is not a multiple of any number in $\pi$. The famous Hall theorem states that a solvable finite group always contains a Hall $\pi$ subgroup and any two Hall $\pi$subgroups of such group are conjugate. The converse of the Hall theorem is also true: for any nonsolvable group $G$, there exists a set $\pi$ such that $G$ does not contain Hall $\pi$subgroups. Nevertheless, Hall $\pi$subgroups may exist in a nonsolvable group. There are examples of sets $\pi$ such that, in any finite group containing a Hall $\pi$subgroup, all Hall $\pi$subgroups are conjugate (and, as a consequence, are isomorphic). In 1987 F. Gross showed that any set $\pi$ of odd primes has this property. In addition, in nonsolvable groups for some sets $\pi$, Hall $\pi$subgroups can be nonconjugate but isomorphic (say, in $PSL_2(7)$ for $\pi=\{2,3\}$) and even nonisomorphic (in $PSL_2(11)$ for $\pi=\{2,3\}$). We prove that the existence of a finite group with nonconjugate Hall $\pi$subgroups for a set $\pi$ implies the existence of a group with nonisomorphic Hall $\pi$subgroups. The converse statement is obvious.
Keywords:
Hall $\pi$subgroup, $\mathscr C_\pi$ condition, conjugate subgroups.
DOI:
https://doi.org/10.21538/0134488920182434350
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UDC:
512.542
MSC: 20D20 Received: 07.05.2018
Citation:
Guo Wen Bin, A. A. Buturlakin, D. O. Revin, “Equivalence of the existence of nonconjugate and nonisomorphic Hall $\pi$subgroups”, Trudy Inst. Mat. i Mekh. UrO RAN, 24, no. 3, 2018, 43–50
Citation in format AMSBIB
\Bibitem{GuoButRev18}
\by Guo~Wen~Bin, A.~A.~Buturlakin, D.~O.~Revin
\paper Equivalence of the existence of nonconjugate and nonisomorphic Hall $\pi$subgroups
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2018
\vol 24
\issue 3
\pages 4350
\mathnet{http://mi.mathnet.ru/timm1549}
\crossref{https://doi.org/10.21538/0134488920182434350}
\elib{http://elibrary.ru/item.asp?id=35511274}
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