Algebraic lattices of solvably saturated formations and their applications

Abstract

In each group G, we select a system of subgroups τ(G) and say that τ is a subgroup functor if G∈τ(G) for every group G, and for every epimorphism φ:A→B and any H∈τ(A) and T∈τ(B), we have Hφ∈τ(B) and Tφ−1∈τ(A). We consider only subgroup functors τ such that for any group G all subgroups of τ(G) are subnormal in G. For any set of groups X, the symbol sτ(X) denotes the set of groups H such that H∈τ(G) for some group G∈X. A formation F is τ-closed if sτ(F)=F. The Frattini subgroup Φ(G) of a group G is the intersection of all maximal subgroups of G. A formation F is said to be solvably saturated if it contains each group G with G/Φ(N)∈F for some solvable normal subgroup N of G. Composition formations are precisely solvably saturated formations. It is shown that the lattice of all τ-closed totally composition formations is algebraic.