Let $\mathfrak F$ be a formation, and let $G$ be a finite group. A subgroup $H$ of $G$ is $\mathrm{K}\mathfrak F$-subnormal (submodular) in $G$ if there is a subgroup chain $H=H_0\le \ H_1 \le \ \ldots \le \ H_{n-1}\le \ H_n=G$ such that for every $i$ either $H_{i}$ is normal in $H_{i+1}$ or $H_{i+1}^\mathfrak{F} \le H_i$ ($H_i$ is a modular subgroup of $H_{i+1}$, respectively). We prove that in a group, a primary subgroup is submodular if and only if it is $\mathrm{K}\mathfrak U_1$-subnormal. Here $\mathfrak U_1$ is a formation of all supersolvable groups with square-free orders of elements. Moreover, for a solvable subgroup-closed formation $\mathfrak{F}$, every solvable $\mathrm{K}\mathfrak{F}$-subnormal subgroup of a group $G$ is contained in the solvable radical of $G$. We also obtain a series of applications of these results to the investigation of groups factorized by $\mathrm{K}\mathfrak{F}$-subnormal and submodular subgroups.
Keywords: finite group, subnormal subgroup, submodular subgroup
Received August 13, 2023
Revised October 6, 2023
Accepted October 9, 2023
Funding Agency: This work was supported by the Belarusian Republican Foundation for Fundamental Research (project no. Φ23PHΦ-237).
Victor Stepanovich Monakhov, Dr. Phys.-Math. Sci., Prof., Francisk Skorina Gomel State University, Gomel, 246028 Belarus, e-mail: victor.monakhov@gmail.com
Irina Leonidovna Sokhor, Cand. Sci. (Phys.-Math.), Francisk Skorina Gomel State University, Gomel, 246028 Belarus, e-mail: irina.sokhor@gmail.com
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Cite this article as: V.S. Monakhov, I.L. Sokhor. On submodularity and $\mathrm{K}\mathfrak F$-subnormality in finite groups. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2023, vol. 29, no. 4, pp. 169–180; Proceedings of the Steklov Institute of Mathematics (Suppl.1), 2023, Vol. 323, Suppl. 1, pp. S168–S178.
