ON TWO PROBLEMS OF THE THEORY OF PARTIALLY TOTALLY COMPOSITION FORMATIONS OF FINITE GROUPS

Abstract

 All groups considered in the paper are assumed to be finite. Further, ω denotes some nonempty set of primes, and τ is a subgroup functor in the sense of A.N. Skiba. Recall that a formation is a class of groups that is closed under taking homomorphic images and finite subdirect products. The paper studies properties of the lattice of all closed functorially totally partially saturated formations. We prove that for any subgroup functor τ, the lattice cτω∞ of all τ-closed totally ω-composition formations is algebraic. Furthermore, we prove that the lattice mentioned above is inductive. In particular, we show that the lattice cτp∞ of all τ-closed totally p-composition formations and the lattice cτ∞ of all τ-closed totally composition formations are both algebraic and inductive. Thus, new classes of algebraic and inductive lattices of formations are found.