Algebraic lattices of solvably saturated formations and their applications

Abstract

In each group G, we select a system of subgroups $$\tau (G)$$ and say that $$\tau$$ is a subgroup functor if $$G \in \tau (G)$$ for every group G, and for every epimorphism $$\varphi : A \rightarrow B$$ and any $$H \in \tau (A)$$ and $$T \in \tau (B)$$ , we have $$H^\varphi \in \tau (B)$$ and $$T^{\varphi ^{-1}} \in \tau (A)$$ . We consider only subgroup functors $$\tau$$ such that for any group G all subgroups of $$\tau (G)$$ are subnormal in G. For any set of groups $${\mathfrak {X}}$$ , the symbol $${s}_\tau ({\mathfrak {X}})$$ denotes the set of groups H such that $$H \in \tau (G)$$ for some group $$G \in {\mathfrak {X}}$$ . A formation $${\mathfrak {F}}$$ is $$\tau$$ -closed if $${s}_\tau ({\mathfrak {F}}) = {\mathfrak {F}}$$ . The Frattini subgroup $${\varPhi }(G)$$ of a group G is the intersection of all maximal subgroups of G. A formation $${\mathfrak {F}}$$ is said to be solvably saturated if it contains each group G with $$G/{\varPhi }(N) \in {\mathfrak {F}}$$ for some solvable normal subgroup N of G. Composition formations are precisely solvably saturated formations. It is shown that the lattice of all $$\tau$$ -closed totally composition formations is algebraic.