On the intersections of the maximal subgroups of finite groups

L. M. Belokon

Problemy fiziki, matematiki i tehniki, no. 4 (2014), pp. 46-59

Résumé

Let $\mathfrak{F}$ be a nonempty radical formation and let $\pi$ be a set of primes. Conditions under which intersections of the maximal subgroups of a finite group mutually simple with numbers from $\pi$ indexes coincide: $\Phi_{\pi,\overline{G_\mathfrak{F}}}(G)=\Phi_\pi(G)$; $\Delta_{\pi,\overline{G_\mathfrak{F}}}^{\mathfrak{F}}(G)=\Delta_{\pi}^{\mathfrak{F}}(G)$; $\overline{\Delta}_{\pi,\overline{G_\mathfrak{F}}}^{\mathfrak{F}}(G)=\Delta_{\pi}^{\mathfrak{F}}(G)$ are investigated. The results following as consequences were established for not necessarily solvable finite groups $G$ on intersections of the maximal subgroups without restrictions on indexes: $\Phi_{\overline{G_\mathfrak{F}}}(G)=\Phi(G)$; $\Delta_{\overline{G_\mathfrak{F}}}^{\mathfrak{F}}(G)=\Delta^{\mathfrak{F}}(G)$; $\overline{\Delta}_{\overline{G_\mathfrak{F}}}^{\mathfrak{F}}(G)=\Delta^{\mathfrak{F}}(G)$. Analogs of statements on intersections $\Phi_\pi(G)$ and $\Delta_\pi^{\mathfrak{F}}(G)$ for not necessarily radical formations are received.

Détail

Mots-clés : radical formations
Keywords: $\mathfrak{F}$-radicals, intersections of maximal subgroups in a finite group.