On the existence of complements for residuals of finite groups

S. F. Kamornikov ; O. L. Shemetkova

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 21 (2015) no. 1, pp. 122-127

Résumé

L.A. Shemetkov's theorem on the complementability of the $\mathfrak{F}$-residual of a finite group is developed in the article. For a local Fitting formation $\mathfrak{F}$, it is proved that, if a group $G$ is representable in the form $G=AB$, where $A$ and $B$ are subnormal subgroups of $G$, the subgroups $A^\mathfrak{F}$ and $B^\mathfrak{F}$ are $\pi(\mathfrak{F})$-solvable and normal in $G$, and Sylow $p$-subgroups of $A^\mathfrak{F}$ and $B^\mathfrak{F}$ are abelian for every $p \in \pi(\mathfrak{F})$, then every $\mathfrak{F}$-normalizer of $G$ is the complement for an $\mathfrak{F}$-residual of $G$.

Détail

Keywords: finite group; subnormal subgroup; formation; residual; complement; local Fitting formation.