On a problem of A. N. Skiba and L. A. Shemetkov concerning Sₙ-closed subformations of a one-generated formation

Throughout this paper, all groups are finite. Let form(G) denote the formation generated by a group G. A formation F is said to be Sₙ-closed if every subnormal subgroup of every group in F belongs to F. A. N. Skiba and L. A. Shemetkov posed the problem of proving that form(G) has only finitely many Sₙ-closed subformations for every group G (Problem 14.81 of the Kourovka Notebook). In this paper we propose a negative answer to this problem. We prove that for the group G₀=(SL₂(5) C₂)₂(5) of order 28800, where C₂ acts on SL₂(5) by an outer automorphism, the formation form(G₀) has an infinite strictly descending chain of Sₙ-closed subformations. The main tool is a family of central-valued multilinear maps, defined for every group in form(G₀) by its outer actions on its components; their vanishing is preserved under taking epimorphic images, subdirect products and subnormal subgroups. The members of the chain are separated by central quotients of the groups SL₂(5)2ᵈ-1 C₂ᵈ, d≥ 2. Using Burichenko's description of the lattice of subformations of form(SL₂(5) C₂), we also prove that every non-empty Sₙ-closed subformation of form(G₀) is generated, as an ordinary formation, by a single group, and that the set of such subformations is countably infinite. Finally, we determine the intersection of the chain and give an explicit generator of it.