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

N. V. Maslova ; D. O. Revin

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 19 (2013) no. 3, pp. 199-206

Résumé

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.

Détail

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