On the properties of the lattice of τ-closed totally ω-composition formations

We study the properties of the lattice c τ ω∞ of all τ -closed totally ω -composition formations of finite groups. We prove the modularity of such a lattice of formations for any subgroup functor τ and any nonempty set ω of primes. In particular, we obtain a positive answer to the question of A. N. Skiba and L. A. Shemetkov (2000) about the modularity of the lattice c ∞ L of all totally L -composition formations. We establish that the lattice c τ ω∞ is a complete sublattice of the lattice c ω ∞ of all totally ω- composition formations of finite groups.