Algebraic lattices of solvably saturated formations and their applications

Abstract

In each group G, we select a system of subgroups sðGÞ and say that s is a subgroup functor if G 2 sðGÞ for every group G, and for every epimorphism u : A ! B and any H 2 sðAÞ and T 2 sðBÞ, we have H u 2 sðBÞ and T u À1 2 sðAÞ. We consider only subgroup functors s such that for any group G all subgroups of sðGÞ are subnormal in G. For any set of groups X, the symbol s s ðXÞ denotes the set of groups H such that H 2 sðGÞ for some group G 2 X. A formation F is s-closed if s s ðFÞ ¼ F. The Frattini subgroup UðGÞ 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=UðNÞ 2 F for some solvable normal subgroup N of G. Composition formations are precisely solvably saturated formations. It is shown that the lattice of all s-closed totally composition formations is algebraic. Keywords Finite group Á Subgroup functor Á Formation of groups Á Satellite of formation Á Totally composition formation Á Algebraic lattice of formations Á Formal language Á Hypergroup Mathematics Subject Classification Primary 20F17 Á Secondary 20D10 Á 43A62 Á 20M35 This work has been supported by the Russian Science Foundation under Grant 18-71-10007.