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 Trudy Inst. Mat. i Mekh. UrO RAN, 2018, Volume 24, Number 3, Pages 43–50 (Mi timm1549)

Equivalence of the existence of nonconjugate and nonisomorphic Hall $\pi$-subgroups

Guo Wen Bina, A. A. Buturlakinbc, D. O. Revinbca

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.

 Funding Agency Grant Number National Natural Science Foundation of China 11771409 Siberian Branch of Russian Academy of Sciences 0314-2016-0001 Russian Foundation for Basic Research 17-51-45025 CAS President's International Fellowship Initiative 2016VMA078 The first author is supported by the National Natural Science Foundation of China (project no. 11771409). The second author is supported by Program I.1.1 for Fundamental Research of the Siberian Branch of the Russian Academy of Sciences (project no. 0314-2016-0001). The third author is supported by CAS President's International Fellowship Initiative (project no. 2016VMA078) and by the Russian Foundation for Basic Research (project no. 17-51-45025).

DOI: https://doi.org/10.21538/0134-4889-2018-24-3-43-50

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Bibliographic databases:

Document Type: Article
UDC: 512.542
MSC: 20D20

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 43--50 \mathnet{http://mi.mathnet.ru/timm1549} \crossref{https://doi.org/10.21538/0134-4889-2018-24-3-43-50} \elib{http://elibrary.ru/item.asp?id=35511274}