ON SOME PROPERTIES OF THE LATTICE OF PARTIALLY TOTALLY SATURATED FORMATIONS OF FINITE GROUPS

Author(s):  V.V. Shcherbina, Belarusian State University, Minsk, Republic of Belarus, shcherbinavv@tut.by

V.G. Safonov, Dr., Prof., Belarusian State University, Minsk, Republic of Belarus

Issue:  Volume 51, № 2

Rubric:  Mathematics

Annotation:  All groups under consideration are finite. The paper studies some properties of the lattice of all -closed totally -saturated formations. We show that for any subgroup functor , the lattice of all -closed totally -saturated formations is modular and algebraic. We also prove that the lattice of all totally -saturated formations is -separable. This strengthens a theorem of V.G. Safonov. Using embeddability the lattice of all -closed totally -saturated formations in the lattice of all totally -saturated formation, we establish that the lattice of all -closed totally -saturated formations is -separable. In particular, we show that the lattice of all -closed totally -saturated formations is modular, algebraic, and -separable as well as the lattice of all -closed totally saturated formations.

Keywords:  formation of finite groups, totally -saturated formation, lattice of formations, -closed formation, modular lattice, algebraic lattice, separable lattice of formations

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