On a problem of A. N. Skiba and L. A. Shemetkov concerning Sn-closed subformations of a one-generated formation
Abstract
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)rtimes C2)×SL2(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−1rtimes C2d, d≥ 2. Using Burichenko's description of the lattice of subformations of form(SL2(5)rtimes 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.
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BibTeX
@misc{lin2026problem,
title = {{On a problem of A. N. Skiba and L. A. Shemetkov concerning $S_n$-closed subformations of a one-generated formation}},
author = {Chenxiao Lin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37462},
eprint = {2610.37462},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37462},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On a problem of A. N. Skiba and L. A. Shemetkov concerning S_n-closed subformations of a one-generated formation
AU - Chenxiao Lin
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37462
UR - https://arxiv.org/abs/2610.37462
AB - 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 S_n-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 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=(SL_2(5)rtimes C_2)×SL_2(5) of order 28800, where C_2 acts on SL_2(5) by an outer automorphism, the formation 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 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 SL_2(5)^2^d−1rtimes C_2^d, d≥ 2. Using Burichenko's description of the lattice of subformations of form(SL_2(5)rtimes C_2), we also prove that every non-empty S_n-closed subformation of 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.
ER -