On the lattices of the $\omega$-fibered formations of finite groups

S. P. Maksakov

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 27 (2021) no. 1, pp. 258-267

Résumé

Only finite groups and classes of finite groups are considered. The lattice approach to the study of formations of groups was first applied by A.N. Skiba in 1986. L.A. Shemetkov and A.N. Skiba established main properties of lattices of local formations and $\omega$-local formations where $\omega$ is a nonempty subset of the set $\mathbb{P}$ of all primes. An $\omega$-local formation is one of types of $\omega$-fibered formations introduced by V.A. Vedernikov and M.M. Sorokina in 1999. Let $f : \omega \cup \{\omega'\} \rightarrow$ $\{$formations of groups$\}$, where $f(\omega') \neq \varnothing$, and $\delta :\mathbb{P} \rightarrow \{$nonempty Fitting formations$\}$ are the functions. Formation $ \frak F = (G \ \vert \ G/O_\omega (G) \in f(\omega')$ and $G/G_{\delta (p)} \in f(p)$ for all $p\in \omega \cap \pi(G) )$ is called an $\omega$-fibered formation with a direction $\delta$ and with an $\omega$-satellite $f$, where $O_{\omega}(G)$ is the largest normal $\omega$-subgroup of the group $G$, $G_{\delta (p)}$ is the $\delta (p)$-radical of the group $G$, i.e. the largest normal subgroup of the group $G$ belonging to the class $\delta (p)$, and $\pi(G)$ is the set of all prime divisors of the order of the group $G$. We study properties of lattices of $\omega$-fibered formations of groups. In this work we have proved the modularity of the lattice $\Theta_{\omega \delta}$ of all $\omega$-fibered formations with the direction $\delta$. Its sublattice $\Theta_{\omega \delta} (\frak F)$ for the definite $\omega$-fibered formation $\frak F$ with the direction $\delta$ is considered. We have established sufficient conditions under which the lattice $\Theta_{\omega \delta} (\frak F)$ is a distributive lattice with complements.

Détail

Keywords: finite group; class of groups; formation; $\omega$-fibered formation; lattice; modular lattice; distributive lattice; lattice with complements.