This article is cited in 2 papers

MATHEMATICS

On commutative semigroups of soluble totally $\omega$-saturated formations

V. G. Safonov, I. N. Safonova

Belarusian State University, Minsk

Abstract: Let $\mathfrak{M}$ be some totally ($n$-multiply) $\omega$-saturated formation of finite groups ($n\geqslant0$), $\mathfrak{F}$ and $\mathfrak{H}$ be totally ($n$-multiply) $\omega$-saturated subformations of $\mathfrak{M}$. Then $A_\infty^\omega(\mathfrak{M})$ ($A_n^\omega(\mathfrak{M})$) denotes the semigroup of all totally ($n$-multiply) $\omega$-saturated subformations of $\mathfrak{M}$ with multiplication $\mathfrak{F}_{\mathfrak{M}}\cdot\mathfrak{H}=\mathfrak{HF}\cap\mathfrak{M}$, where $\mathfrak{HF}=(G|G^{\mathfrak{H}}\in\mathfrak{F})$. It is proved that a soluble totally ($n$-multiply) $\omega$-saturated formation generates a commutative semigroup of totally ($n$-multiply) $\omega$-saturated subformations if and only if, when it is nilpotent. In particular, the problem 6.26 from [1] is solved for the class of soluble groups.

Keywords: formation of finite groups, totally $\omega$-saturated formation, $n$-multiply $\omega$-saturated formation, semigroup of formations, commutative semigroup of formation.

UDC: 512.542

Received: 26.10.2015