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

Abstract

Throughout this paper, all groups are finite. Let $\mathrm{form}(G)$ denote the formation generated by a group $G$. A formation $\mathfrak{F}$ is said to be $S_n$-closed if every subnormal subgroup of every group in $\mathfrak{F}$ belongs to $\mathfrak{F}$. A. N. Skiba and L. A. Shemetkov posed the problem of proving that $\mathrm{form}(G)$ has only finitely many $S_n$-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 $G_0=(\mathrm{SL}_2(5)\rtimes C_2)\times\mathrm{SL}_2(5)$ of order 28800, where $C_2$ acts on $\mathrm{SL}_2(5)$ by an outer automorphism, the formation $\mathrm{form}(G_0)$ has an infinite strictly descending chain of $S_n$-closed subformations. The main tool is a family of central-valued multilinear maps, defined for every group in $\mathrm{form}(G_0)$ 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 $\mathrm{SL}_2(5)^{2^d-1}\rtimes C_2^d$, $d\geq 2$. Using Burichenko's description of the lattice of subformations of $\mathrm{form}(\mathrm{SL}_2(5)\rtimes C_2)$, we also prove that every non-empty $S_n$-closed subformation of $\mathrm{form}(G_0)$ 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.