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A. F. Vasil'ev

Publications and source records attributed to A. F. Vasil'ev.

6 recordsLinked to original sources
arXiv2024

On $\mathrm{K}$-$\mathbb{P}_{t}$-subnormal subgroups of finite groups and related formations

Let $t$ be a fixed natural number. A subgroup $H$ of a group $G$ will be called $\mathrm{K}$-$\mathbb{P}_{t}$-subnormal in $G$ if there exists a chain of subgroups $H = H_{0} \leq H_{1} \leq \cdots \leq H_{m-1} \leq H_{m} = G$ such that either $H_{i-1}$ is normal in $H_{i}$ or $|H_{i} : H_{i-1}|$ is a some prime $p$ and $p-1$ is not divisible by the $(t+1)$th powers of primes for every $i = 1,\ldots , n$. In this work, properties of $\mathrm{K}$-$\mathbb{P}_{t}$-subnormal subgroups and classes of groups with Sylow $\mathrm{K}$-$\mathbb{P}_{t}$-subnormal subgroups are obtained.

math.GR↗
arXiv2019

Finite groups with $\mathfrak{F}$-subnormal normalizers of Sylow subgroups

Let $π$ be a set of primes and $\mathfrak{F}$ be a formation. In this article a properties of the class ${\rm w}^{*}_π\mathfrak{F}$ of all groups $G$, such that $π(G)\subseteq π(\mathfrak{F})$ and the normalizers of all Sylow $p$-subgroups of $G$ are $\mathfrak{F}$-subnormal in $G$ for every $p\inπ\capπ(G)$ are investigated. It is established that ${\rm w}^{*}_π\mathfrak{F}$ is a formation. Some hereditary saturated formations $\mathfrak{F}$ for which ${\rm w}^{*}_π\mathfrak{F}=\mathfrak{F}$ are founded.

math.GR↗
arXiv2015

Arithmetic graphs of finite groups

In this paper we introduced an arithmetic graph function which associates with every group G the directed graph whose vertices corresponds to the divisors of |G|. With the help of such functions we introduced arithmetic graphs of classes of groups, in particular of hereditary saturated formations. We formulated the problem of the recognition of classes of groups by arithmetic graph functions and investigated this problem for some arithmetic graph functions.

math.GR↗
arXiv2013

Generalized Fitting subgroups of finite groups

In this paper we consider the Fitting subgroup $F(G)$ of a finite group $G$ and its generalizations: the quasinilpotent radical $F^*(G)$ and the generalized Fitting subgroup $\tilde{F}(G)$ defined by $\tilde{F}(G)\supseteq Φ(G)$ and $\tilde{F}(G)/Φ(G)=Soc(G/Φ(G))$. We sum up known properties of $\tilde{F}(G)$ and suggest some new ones. Let $R$ be a subgroup of a group $G$. We shall call a subgroup $H$ of $G$ the $R$-subnormal subgroup if $H$ is subnormal in $ \langle H,R\rangle$. In this work the influence of $R$-subnormal subgroups (maximal, Sylow, cyclic primary) on the structure of finite groups are studied in the case when $R\in\{F(G), F^*(G),\tilde{F}(G)\}$.

math.GR↗
arXiv2012

On partially conjugate-permutable subgroups of finite groups

Let $R$ be a subset of a group $G$. We call a subgroup $H$ of $G$ the $R$-conjugate-permutable subgroup of $G$, if $HH^{x}=H^{x}H$ for all $x\in R$. This concept is a generalization of conjugate-permutable subgroups introduced by T. Foguel. Our work focuses on the influence of $R$-conjugate-permutable subgroups on the structure of finite groups in case when $R$ is the Fitting subgroup or its generalizations $F^{*}(G)$ (introduced by H. Bender in 1970) and $\tilde{F}(G)$ (introduced by P. Shmid 1972). We obtain a new criteria for nilpotency and supersolubility of finite groups which generalize some well known results.

math.GR↗