MATHEMATICS

On reducible $\tau$-closed $\omega$-saturated formations with a soluble defect 2

V. G. Safonov^a, I. N. Safonova<sup>b

a</sup> Ministry of Education of the Republic of Belarus, Minsk
^b F. Skorina Gomel State University, Gomel

Abstract: Let $\mathfrak F$ be some $\tau$-closed $\omega$-saturated formation, $\mathfrak S$ be the formation of all soluble groups. Then $\mathfrak F/^{\omega}_{\tau}\mathfrak F \cap \mathfrak S$ denotes the lattice of all $\tau$-closed $\omega$-saturated formations $\mathfrak H$ such that $\mathfrak F \cap \mathfrak S \subseteq \mathfrak H \subseteq \mathfrak F$. A length of the lattice $\mathfrak F/^{\omega}_{\tau}\mathfrak F \cap \mathfrak S$ is called a soluble $l^{\omega}_{\tau}$-defect of the $\tau$-closed $\omega$-saturated formation $\mathfrak F$. The description of reducible $\tau$-closed $\omega$-saturated formations of finite groups with a soluble $l^{\omega}{\tau}$-defect 2 is obtained.

Keywords: formation of finite groups, $\omega$-saturated formation, defect of a formation, lattice of formations, $\tau$-closed formation.

UDC: 519.542

Received: 23.11.2009