Soluble formations with the Shemetkov property

V. I. Murashka

Problemy fiziki, matematiki i tehniki, no. 1 (2015), pp. 82-87

Résumé

All saturated soluble formations whose all $s$-critical groups are soluble were described. With every local formation $\mathfrak{F}=LF(f)$, such that $f(p)=\mathfrak{S}_{\pi(f(p))}$ for all $p\in\pi(\mathfrak{F})$ and $f(p)=\varnothing$ otherwise, was associated directed graph $\Gamma(\mathfrak{F},f)$ without loops whose vertices are prime numbers from $\pi(\mathfrak{F})$ and $(p_i,p_j)$ is an edge of $\Gamma(\mathfrak{F},f)$ if and only if $p_j\in\pi(f(p_i))$. With the help of such kind’s graphs all hereditary soluble formations with the Shemetkov property were described.

Détail

Keywords: minimal simple group, $s$-critical group, hereditary local formation, formation with the Shemetkov property, graph associated with formation.