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Viachaslau I. Murashka

Publications and source records attributed to Viachaslau I. Murashka.

16 recordsLinked to original sources
arXiv2026

On the intersection of $\mathfrak{F}$-maximal subgroups of a finite group

We investigate the properties of the intersection $\mathrm{Int}_{\mathfrak{F}}(G)$ of all $\mathfrak{F}$-maximal subgroups of a finite group $G$ for a hereditary formation $\mathfrak{F}$ of finite groups. We prove that $\mathrm{Int}_{\mathfrak{F}}(G/\mathrm{Int}_{\mathfrak{F}}(G))\simeq 1$ holds for any finite group $G$ if and only if $\mathfrak{F}$ contains every group $G$ all of whose $\mathfrak{F}$-subgroups are $\mathfrak{F}$-subnormal. As corollaries we obtain the results of A. N. Skiba (2011), J. C. Beidleman and H. Heineken (2011) about $\mathrm{Int}_{\mathfrak{F}}(G)$ for a hereditary saturated formation $\mathfrak{F}$.

math.GR↗
arXiv2025

On the Length of a Maximal Subgroup of a Finite Group

For a finite group $G$ and its maximal subgroup $M$ we proved that the generalized Fitting height of $M$ can't be less by 2 than the generalized Fitting height of $G$ and the non-$p$-soluble length of $M$ can't be less by 1 than the non-$p$-soluble length of $G$. We constructed a hereditary saturated formation $\mathfrak{F}$ such that $\{n_σ(G, \mathfrak{F})-n_σ(M, \mathfrak{F})\mid G$ is finite $σ$-soluble and $M$ is a maximal subgroup of $G\}=\mathbb{N}\cup\{0\}$ where $n_σ(G, \mathfrak{F})$ denotes the $σ$-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.

math.GR↗
arXiv2024

Formations of Finite Groups in Polynomial Time: the $\mathfrak{F}$-Hypercenter

For a wide family of formations $\mathfrak{F}$ (which includes Baer-local formations) it is proved that the $ \mathfrak{F}$-hypercenter of a permutation finite group can be computed in polynomial time. In particular, the algorithms for computing the $\mathfrak{F}$-hypercenter for the following classes of groups are suggested: hereditary local formations with the Shemetkov property, rank formations, formations of all quasinilpotent, Sylow tower, $p$-nilpotent, supersoluble, $w$-supersoluble and $SC$-groups. For some of these formations algorithms for the computation of the intersection of all maximal $\mathfrak{F}$-subgroups are suggested.

math.GR↗
arXiv2024

Formations of Finite Groups in Polynomial Time: the $\mathfrak{F}$-Radical

For a Baer-local (composition) Fitting formation $\mathfrak{F}$ the polynomial time algorithm for the computation of the $\mathfrak{F}$-radical of a permutation group is suggested. In particular it is showed how one can compute the $\mathfrak{F}$-radical in case when $\mathfrak{F}$ is a primitive saturated formation of soluble groups. Moreover, the polynomial time algorithms for the computation of different lengthes associated with a group are presented.

math.GR↗
arXiv2024

Constructing regular saturated formations of finite soluble groups

For a formation $\mathfrak{F}$ of finite groups consider a graph whose vertices are elements of a finite group and two vertices are connected by an edge if and only if they generates non-$\mathfrak{F}$-group as elements of a group. A hereditary formation $\mathfrak{F}$ is called regular if the set of all isolated vertices of the described graph coincides with the intersection of all maximal $\mathfrak{F}$-subgroups in every group. The constructive description of saturated regular formations of soluble groups which improves the results of Lucchini and Nemmi is obtained in this paper. In particular, it is showed that saturated regular non-empty formations of soluble groups are just hereditary formations $\mathfrak{F}$ of soluble groups that contains every group all whose cyclic primary subgroups are $K$-$\mathfrak{F}$-subnormal. Also we prove that the lattice of saturated regular formations of soluble groups is lattice isomorphic to the Steinitz's lattice.

math.GR↗
arXiv2024

$σ$-properties of finite groups in polynomial time

Let $H, K$ be subgroups of the permutation group $G$ of degree $n$ with $K\trianglelefteq G$ and $σ$ be a partition of the set of all different prime divisors of $|G/K|$. We prove that in polynomial time (in $n$) one can check $G/K$ for $σ$-nilpotency and $σ$-solubility; $H/K$ for $σ$-subnormality and $σ$-$p$-permutability in $G/K$. Moreover one can find the least partition $σ$ of $π(G/K)$ for which $G/K$ is $σ$-nilpotent. Also one can find the least partition $σ$ of $π(G/K)$ for which $H/K$ is $σ$-$p$-permutable in $G/K$.

math.GR↗
arXiv2023

Arithmetic graphs and the products of finite groups

The Hawkes graph $Γ_H(G)$ of $G$ is the directed graph whose vertex set coincides with $π(G)$ and it has the edge $(p, q)$ whenever $q\inπ(G/O_{p',p}(G))$. The Sylow graph $Γ_s(G)$ of $G$ is the directed graph with vertex set $π(G)$ and $(p, q)$ is an edge of $Γ_s(G)$ whenever $q \inπ(N_G(P)/PC_G(P))$ for some Sylow $p$-subgroup $P$ of $G$. The $N$-critical graph $Γ_{Nc}(G)$ of a group $G$ the directed graph whose vertex set coincides with $π(G)$ such that $(p, q)$ is an edge of $Γ_{Nc}(G)$ whenever $G$ contains a Schmidt $(p, q)$-subgroup, i.e. a Schmidt $\{p, q\}$-subgroup with a normal Sylow $p$-subgroup. In the paper the Hawkes, the Sylow and the $N$-critical graphs of the products of totally permutable, mutually permutable and $\mathfrak{N}$-connected subgroups are studied.

math.GR↗
arXiv2023

On the Generalized Fitting Height and Nonsoluble Length of the Mutually Permutable Products of Finite Groups

The generalized Fitting height $h^*(G)$ of a finite group $G$ is the least number $h$ such that $\mathrm{F}_h^* (G) = G$, where $\mathrm{F}_{(0)}^* (G) = 1$, and $\mathrm{F}_{(i+1)}^*(G)$ is the inverse image of the generalized Fitting subgroup $\mathrm{F}^*(G/\mathrm{F}^*_{(i)} (G))$. Let $p$ be a prime, $1=G_0\leq G_1\leq\dots\leq G_{2h+1}=G$ be the shortest normal series in which for $i$ odd the factor $G_{i+1}/G_i$ is $p$-soluble (possibly trivial), and for $i$ even the factor $G_{i+1}/G_i$ is a (non-empty) direct product of nonabelian simple groups. Then $h=λ_p(G)$ is called the non-$p$-soluble length of a group $G$. We proved that if a finite group $G$ is a mutually permutable product of of subgroups $A$ and $B$ then $\max\{h^*(A), h^*(B)\}\leq h^*(G)\leq \max\{h^*(A), h^*(B)\}+1$ and $\max\{λ_p(A), λ_p(B)\}= λ_p(G)$. Also we introduced and studied the non-Frattini length.

math.GR↗
arXiv2021

A generalization of Hall's theorem on hypercenter

Let $ σ$ be a partition of the set of all primes and $\mathfrak{F}$ be a hereditary formation. We described all formations $\mathfrak{F}$ for which the $\mathfrak{F}$-hypercenter and the intersection of weak $K$-$\mathfrak{F}$-subnormalizers of all Sylow subgroups coincide in every group. In particular the formation of all $σ$-nilpotent groups has this property. With the help of our results we solve a particular case of L.A.~Shemetkov's problem about the intersection of $\mathfrak{F}$-maximal subgroups and the $\mathfrak{F}$-hypercenter. As corollaries we obtained P. Hall's and R. Baer's classical results about the hypercenter. We proved that the non-$σ$-nilpotent graph of a group is connected and its diameter is at most 3.

math.GR↗
arXiv2021

On ECP-Groups

According to T. Foguel a subgroup $H$ of a group $G$ is called conjugate-permutable if $ HH^x=H^xH$ for every $x\in G$. Mingyao Xu and Qinhai Zhang studied finite groups with every subgroup conjugate-permutable (ECP-groups) and asked three questions about them. We gave the answers on these questions. In particular, every group of exponent 3 is ECP-group, there exist non-regular ECP-$3$-groups and the class of all finite ECP-groups is neither formation nor variety.

math.GR↗
arXiv2021

Fitting like subgroups of finite groups

In this paper the concept of $\mathbb{F}$-functorial of a finite group was introduced. These functorials have many properties of the Fitting subgroup of a soluble group and the generalized Fitting subgroup of a finite group. It was shown that the set of all $\mathbb{F}$-functorials is a complete distributive lattice and the cardinality of this lattice is continuum. The sharp bounds on the generalized Fitting height of mutually permutable product of two subgroups were obtained.

math.GR↗
arXiv2020

On a generalization of one Kramer's theorem

Yangming Li and Xianhua Li in 2012 proposed a conjecture that generalizes O.U. Kramer's result about supersoluble groups. Here we proved that this conjecture is false in the general case and true for groups with the trivial Frattini subgroup.

math.GR↗
arXiv2020

On some applications of Fitting like subgroups of finite groups

In this paper we study the groups all whose maximal or all Sylow subgroups are $K$-$\mathfrak{F}$-subnormal in their product the with generalizations of the Fitting subgroup $\mathrm{F}^*(G)$ and $\mathrm{\tilde F}(G)$. We prove that a hereditary formation $\mathfrak{F}$ contains every group all whose Sylow subgroups are $K$-$\mathfrak{F}$-subnormal in their product with $\mathrm{F}^*(G)$ if and only if $\mathfrak{F}$ is the class of all $σ$-nilpotent groups for some partition $σ$ of the set of all primes. We obtain a new characterization of the $σ$-nilpotent hypercenter, i.e. the $\mathfrak{F}$-hypercenter and the normal largest subgroup which $K$-$\mathfrak{F}$-subnormalize all Sylow subgroups coincide if and only if $\mathfrak{F}$ is the class of all $σ$-nilpotent groups.

math.GR↗
arXiv2019

On one question of Shemetkov about composition formations

In this paper one construction of composition formations was introduced. This construction contains formations of quasinilpotent groups, $c$-supersoluble groups, groups defined by ranks of chief factors and some new classes of groups. A partial answer on a question of L.\,A. Shemetkov about the intersection of $\mathfrak{F}$-maximal subgroups and the $\mathfrak{F}$-hypercenter was given for these composition formations.

math.GR↗
arXiv2013

One formation of finite groups

It is shown that two formations of finite groups, one was introduced by V.S. Monakhov and V.N. Kniahina and another one was introduced by R. Brandl, are coincides.

math.GR↗