Separability of the lattice of $\tau$-closed totally $\omega$-saturated formations of finite groups

V. G. Safonov ; I. N. Safonova

Problemy fiziki, matematiki i tehniki, no. 4 (2017), pp. 76-83

Résumé

Let $\mathfrak{X}$ be a non-empty class of finite groups. A complete lattice $\theta$ of formations is said to be $\mathfrak{X}$-separable if for every term $\nu(x_1,\dots, x_n)$ of signature $\{\cap,\lor_\theta\}$, $\theta$-formations $\mathfrak{F}_1,\dots,\mathfrak{F}_n$ and every group $A\in\mathfrak{X}\cap\nu(\mathfrak{F}_1,\dots,\mathfrak{F}_n)$ exists $\mathfrak{X}$-groups $A_1\in \mathfrak{F}_1,\dots, A_n\in\mathfrak{F}_n$, such that $A\in\nu(\theta\mathrm{form}A_1, \dots, \theta\mathrm{form}A_n)$. In particular, if $\mathfrak{X}=\mathfrak{G}$ is the class of all finite groups then the lattice $\theta$ of formations is said to be $\mathfrak{G}$-separable or, briefly, separable. It is proved that the lattice $l^\tau_{\omega_{\infty}}$ of all $\tau$-closed totally $\omega$-saturated formations is $\mathfrak{G}$-separable for any subgroup functor $\tau$.

Détail

Keywords: formation of finite groups, $\tau$-closed formation, totally $\omega$-saturated formation, lattice of formations, $\mathfrak{G}$-separated lattice of formations.