arXiv subjects

Danila Revin

Publications and source records attributed to Danila Revin.

5 recordsLinked to original sources

arXiv2019

The proper definition and Wielandt-Hartley's theorem for submaximal $\mathfrak{X}$-subgroups

A nonempty class $\mathfrak{X}$ of finite groups is called complete if it is closed under taking subgroups, homomorphic images and extensions. We deal with a classical problem of determining $\mathfrak{X}$-maximal subgroups. We consider two definitions of submaximal $\mathfrak{X}$-subgroups suggested by Wielandt and discuss which one better suits our task. We prove that these definitions are not equivalent yet Wielandt-Hartley's theorem holds true for either definition of $\mathfrak{X}$-submaximality. We also give some applications of the strong version of Wielandt-Hartley's theorem.

math.GR↗

arXiv2017

Groups with bounded centralizer chains and the~Borovik--Khukhro conjecture

Let $G$ be a locally finite group and $F(G)$ the Hirsch--Plotkin radical of $G$. Denote by $S$ the full inverse image of the generalized Fitting subgroup of $G/F(G)$ in $G$. Assume that there is a number $k$ such that the length of every chain of nested centralizers in $G$ does not exceed $k$. The Borovik--Khukhro conjecture states, in particular, that under this assumption the quotient $G/S$ contains an abelian subgroup of index bounded in terms of $k$. We disprove this statement and prove some its weaker analog.

math.GR↗

arXiv2017

Classification and properties of the $π$-submaximal subgroups in minimal nonsolvable groups

Let $π$ be a set of primes. According to H. Wielandt, a subgroup $H$ of a finite group $X$ is called a $π$-submaximal subgroup if there is a monomorphism $ϕ:X\rightarrow Y$ into a finite group $Y$ such that $X^ϕ$ is subnormal in $Y$ and $H^ϕ=K\cap X^ϕ$ for a $π$-maximal subgroup $K$ of $Y$. In his talk at the well-known conference on finite groups in Santa-Cruz (USA) in 1979, Wielandt posed a series of open questions and among them the following problem: to describe the $π$-submaximal subgroup of the minimal nonsolvable groups and to study properties of such subgroups: the pronormality, the intravariancy, the conjugacy in the automorphism group etc. In the article, for every set $π$ of primes, we obtain a description of the $π$-submaximal subgroup in minimal nonsolvable groups and investigate their properties, so we give a solution of Wielandt's problem.

math.GR↗

arXiv2014

Confirmation for Wielandt's conjecture

Let $π$ be a set of primes. By H.Wielandt definition, {\it Sylow $π$-theorem} holds for a finite group $G$ if all maximal $π$-subgroups of $G$ are conjugate. In the paper, the following statement is proven. Assume that $π$ is a union of disjoint subsets $σ$ and $τ$ and a finite group $G$ possesses a $π$-Hall subgroup which is a direct product of a $σ$-subgroup and a $τ$-subgroup. Furthermore, assume that both the Sylow $σ$-theorem and $τ$-theorem hold for $G$. Then the Sylow $π$-theorem holds for $G$. This result confirms a conjecture posed by H.\,Wielandt in~1959.

math.GR↗

arXiv2014

Frattini Argument for Hall subgroups

In the paper, it is proved that if a finite group $G$ possesses a $π$-Hall subgroup for a set $π$ of primes, then every normal subgroup $A$ of $G$ possesses a $π$-Hall subgroup $H$ such that ${G=AN_G(H)}$.

math.GR↗