mathnasТруды Института математики НАН БеларусиProceedings of the Institute of Mathematics of the NAS of Belarus1812-5093Институт математики НАН Беларусиmathnas-66Research ArticleАЛГЕБРА И ТЕОРИЯ ЧИСЕЛALGEBRA AND NUMBER THEORYК теореме К. ДёркаTo the theorem of K. DoerkМурашкоВ. И.MurashkaV. I.

Гомель

Gomel

mvimath@yandex.ruВасильевА. Ф.Vasil’evA. F.

Гомель

Gomel

ormation56@mail.ruГомельский государственный университет им. Ф. СкориныFrancisk Skorina Gomel State University2025300620253312833Copyright © Мурашко В.И., Васильев А.Ф., 20252025Мурашко В.И., Васильев А.Ф.Murashka V.I., Vasil’ev A.F.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://mathnas.ejournal.by/jour/article/view/66

Для конечной группы $G$ и ее максимальной подгруппы $M$ мы доказали, что обобщенная высота Фиттинга группы $G$ минус  обобщенная высота Фиттинга подгруппы $M$ не превосходит 2, а не-$p$-разрешимая длина группы $G$ минус не-$p$-разрешимая длина подгруппы $M$ не превосходит 1. Мы построили наследственную насыщенную формацию $\mathfrak{F}$ так, что $\{n_\sigma(G, \mathfrak{F})-n_\sigma(M, \mathfrak{F})\mid G$ конечна $\sigma$-разрешима и $M$ является максимальной подгруппой группы $G\}=\mathbb{N}\cup\{0\}$, где $n_\sigma(G, \mathfrak{F})$ обозначает $\sigma$-нильпотентную длину $\mathfrak{F}$-корадикала группы $G$. Эта конструкция показывает, что результаты об обобщенных длинах максимальных подгрупп, опубликованные в Math. Nachr. (1994) и Mathematics (2020), являются некорректными.

For a finite group $G$ and its maximal subgroup $M$ we proved that the generalized Fitting height of $G$ minus the generalized Fitting height of $M$ is not greater than 2 and the non-$p$-soluble length of $G$ minus the non-$p$-soluble length of $M$ is not greater than 1. We constructed a hereditary saturated formation $\mathfrak{F}$ such that $\{n_\sigma(G, \mathfrak{F})-n_\sigma(M, \mathfrak{F})\mid G$ is finite $\sigma$-soluble and $M$ is a maximal subgroup of $G\}=\mathbb{N}\cup\{0\}$ where $n_\sigma(G, \mathfrak{F})$ denotes the $\sigma$-nilpotent length of the $\mathfrak{F}$-residual of  $G$.

This construction shows the results about the generalized lengths of maximal subgroups published in Math. Nachr. (1994) and Mathematics (2020) are not correct.

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The authors declare that there are no conflicts of interest present.