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 Trudy Inst. Mat. i Mekh. UrO RAN, 2013, Volume 19, Number 3, Pages 199–206 (Mi timm977)

Generation of a finite group with Hall maximal subgroups by a pair of conjugate elements

N. V. Maslovaab, D. O. Revincd

a Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences
b Ural Federal University named B. N. Yeltsin
c Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences
d Novosibirsk State University

Abstract: For a finite group $G$, the set of all prime divisors of $|G|$ is denoted by $\pi(G)$. P. Shumyatskii introduced the following conjecture, which is included in the “Kourovka Notebook” as Question 17.125: a finite group $G$ always contains a pair of conjugate elements $a$ and $b$ such that $\pi(G)=\pi(\langle a,b\rangle)$. Denote by $\mathfrak Y$ the class of all finite groups $G$ such that $\pi(H)\ne\pi(G)$ for every maximal subgroup $H$ in $G$. Shumyatskii's conjecture is equivalent to the following conjecture: every group from $\mathfrak Y$ is generated by two conjugate elements. Let $\mathfrak V$ be the class of all finite groups in which every maximal subgroup is a Hall subgroup. It is clear that $\mathfrak V\subseteq\mathfrak Y$. We prove that every group from $\mathfrak V$ is generated by two conjugate elements. Thus, Shumyatskii's conjecture is partially supported. In addition, we study some properties of a smallest order counterexample to Shumyatskii's conjecture.

Keywords: finite group, generation by a pair of conjugate elements, Hall subgroup, maximal subgroup, prime spectrum.

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English version:
Proceedings of the Steklov Institute of Mathematics (Supplementary issues), 2014, 285, suppl. 1, S139–S145

Bibliographic databases:

UDC: 512.542

Citation: N. V. Maslova, D. O. Revin, “Generation of a finite group with Hall maximal subgroups by a pair of conjugate elements”, Trudy Inst. Mat. i Mekh. UrO RAN, 19, no. 3, 2013, 199–206; Proc. Steklov Inst. Math. (Suppl.), 285, suppl. 1 (2014), S139–S145

Citation in format AMSBIB
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This publication is cited in the following articles:
1. N. V. Maslova, D. O. Revin, “On nonabelian composition factors of a finite group that is prime spectrum minimal”, Proc. Steklov Inst. Math. (Suppl.), 287, suppl. 1 (2014), 116–127
2. N. V. Maslova, “Finite simple groups that are not spectrum critical”, Proc. Steklov Inst. Math. (Suppl.), 292, suppl. 1 (2016), 211–215
3. N. V. Maslova, “Finite groups with arithmetic restrictions on maximal subgroups”, Algebra and Logic, 54:1 (2015), 65–69
4. N. V. Maslova, “On the finite prime spectrum minimal groups”, Proc. Steklov Inst. Math. (Suppl.), 295, suppl. 1 (2016), 109–119
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