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Evgeny P. Vdovin

Publications and source records attributed to Evgeny P. Vdovin.

9 recordsLinked to original sources
arXiv2021

On the sharp Baer--Suzuki theorem for the $π$-radical

Let $π$ be a set of primes such that $|π|\geqslant 2$ and $π$ differs from the set of all primes. Denote by $r$ the smallest prime which does not belong to $π$ and set $m=r$ if $r=2,3$ and $m=r-1$ if $r\geqslant 5$. We study the following conjecture: a conjugacy class $D$ of a finite group $G$ is contained in $Oπ(G)$ if and only if every $m$ elements of $D$ generate a $π$-subgroup. We confirm this conjecture for each group $G$ whose nonabelian composition factors are isomorphic to alternating, linear and unitary simple groups.

math.GR↗
arXiv2021

Intersection of Solvable Hall subgroups in finite groups

In this paper we show that if S is a simple classical group, a group G is contained in inner-diagonal automorphisms of S and contains S, and H is a solvable Hall subgroup of G, then there exists five conjugates of H, whose intersection is trivial.

math.GR↗
arXiv2020

The reduction theorem for relatively maximal subgroups

Let $\mathfrak{X}$ be a class of finite groups closed under taking subgroups, homomorphic images and extensions. It is known that if $A$ is a normal subgroup of a finite group $G$ then the image of an $\mathfrak{X}$-maximal subgroup $H$ of $G$ in $G/A$ is not, in general, $\mathfrak{X}$-maximal in $G/A$. We say that the reduction $\mathfrak{X}$-theorem holds for a finite group $A$ if, for every finite group $G$ that is an extension of $A$ (i. e. contains $A$ as a normal subgroup), the number of conjugacy classes of $\mathfrak{X}$-maximal subgroups in $G$ and $G/A$ is the same. The reduction $\mathfrak{X}$-theorem for $A$ implies that $HA/A$ is $\mathfrak{X}$-maximal in $G/A$ for every extension $G$ of $A$ and every $\mathfrak{X}$-maximal subgroup $H$ of $G$. In this paper, we prove that the reduction $\mathfrak{X}$-theorem holds for $A$ if and only if all $\mathfrak{X}$-maximal subgroups are conjugate in $A$ and classify the finite groups with this property in terms of composition factors.

math.GR↗
arXiv2019

Baer--Suzuki theorem for the $π$-radical

In the paper we prove (modulo the classification of finite simple groups) an analogue of the famous Baer-Suzuki theorem for the $π$-radical of a finite group, where $π$ is a set of primes

math.GR↗
arXiv2018

The Hall property $\mathcal{D}_π$ is inherited by overgroups of $π$-Hall subgroups

Let $π$ be a set of primes. We say that a finite group $G$ is a $\mathcal{D}_π$-group if the maximal $π$-subgroups of $G$ are conjugate. In this paper, we give an affirmative answer to Problem 17.44(b) from "Kourovka notebook", namely we prove that in a $\mathcal{D}_π$-group an overgroup of a $π$-Hall subgroup is always a $\mathcal{D}_π$-group.

math.GR↗
arXiv2013

On the pronormality of Hall subgroups

Fix a set of primes $π$. A finite group is said to satisfy $C_π$ or, in other words, to be a $C_π$-group, if it possesses exactly one class of conjugate $π$-Hall subgroups. The pronormality of $π$-Hall subgroups in $C_π$-groups is proven, or, equivalently, we prove that $C_π$ is inherited by overgroups of $π$-Hall subgroups. Thus an affirmative solution to Problem 17.44(a) from the "Kourovka notebook" is obtained. We also provide an example, showing that Hall subgroups in finite groups are not pronormal in general.

math.GR↗
arXiv2010

An adjacency criterion for the prime graph of a finite simple group

In the paper we give an exhaustive arithmetic criterion of adjacency in prime graph $GK(G)$ for every finite nonabelian simple group $G$. By using this criterion for all finite simple groups an independence set with the maximal number of vertices, an independence set containing 2 with the maximal number of vertices, and the orders of these independence sets are given. We assemble this information in the tables at the end of the paper. Several applications of obtained results for various problems of finite group theory are considered.

math.GR↗