W. Guo, A.A. Buturlakin, D.O. Revin. Equivalence of the existence of nonconjugate and nonisomorphic Hall $\pi$-subgroups

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.

The paper was received by the Editorial Office on May 7, 2018.

Funding Agency: 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).

Wenbin Guo, Dr. Phys.-Math. Sci, Prof., University of Science and Technology of China, Hefei, 230026 China, e-mail: wbguo@ustc.edu.cn.

Aleksandr Aleksandrovich Buturlakin, Cand. Sci (Phys.-Math.), Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, 630090 Russia; Novosibirsk State University, Novosibirsk, 630090 Russia, e-mail: buturlakin@math.nsc.ru.

Danila Olegovich Revin, Dr. Phys.-Math. Sci, Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, 630090 Russia; Novosibirsk State University, Novosibirsk, 630090 Russia; University of Science and Technology of China, Hefei, 230026 China, e-mail: revin@math.nsc.ru.


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