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Alexander N. Skiba

Publications and source records attributed to Alexander N. Skiba.

At least 19 recordsLinked to original sources
arXiv2024

A contribution to the theory of $σ$-properties of a finite group

We characterize some classes of finite soluble groups. In particular, we prove that: a finite group $G$ is supersoluble if and only if $G$ has a normal subgroup $D$ such that $G/D$ is supersoluble and $D$ avoids every chief factor of $G$ between $V^{G}$ and $V_{G}$ for every maximal subgroup $V$ of the generalized Fitting subgroup $F^{*}(G)$ of $G$; a finite soluble group $G$ is a $PST$-group (that is, Sylow permutability is a transitive relation on $G$) if and only if $G$ has a normal subgroup $D$ such that $G/D$ is nilpotent and $D$ avoids every chief factor of $G$ between $V^{G}$ and $V_{G}$ for every subnormal subgroup $A$ of $G$.

math.GR↗
arXiv2021

A $G$-covering subgroup system of a finite group for some classes of $σ$-soluble groups

Let ${\frak F}$ be a class of group and $G$ a finite group. Then a set $Σ$ of subgroups of $G$ is called a \emph{$G$-covering subgroup system} for the class ${\frak F}$ if $G\in {\frak F}$ whenever $Σ\subseteq {\frak F}$. We prove that: {\sl If a set of subgroups $Σ$ of $G$ contains at least one supplement to each maximal subgroup of every Sylow subgroup of $G$, then $Σ$ is a $G$-covering subgroup system for the classes of all $σ$-soluble and all $σ$-nilpotent groups, and for the class of all $σ$-soluble $PσT$-groups.} This result gives positive answers to questions 19.87 and 19.88 from the Kourovka notebook.

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arXiv2020

A generalisation of Schenkman's theorem

Let $G$ be a finite group and let $\mathfrak{F}$ be a hereditary saturated formation. We denote by $\mathbf{Z}_{\mathfrak{F}}(G)$ the product of all normal subgroups $N$ of $G$ such that every chief factor $H/K$ of $G$ below $N$ is $\mathfrak{F}$-central in $G$, that is, \[ (H/K) \rtimes (G/\mathbf{C}_{G}(H/K)) \in \mathfrak{F}. \]A subgroup $A \leq G$ is said to be $\mathfrak{F}$-subnormal in the sense of Kegel, or $K$-$\mathfrak{F}$-subnormal in $G$, if there is a subgroup chain \[ A = A_0 \leq A_1 \leq \ldots \leq A_n = G \] such that either $A_{i-1} \trianglelefteq A_{i}$ or $A_i / (A_{i-1})_{A_i} \in \mathfrak{F}$ for all $i = 1, \ldots , n$. In this paper, we prove the following generalisation of Schenkman's Theorem on the centraliser of the nilpotent residual of a subnormal subgroup: Let $\mathfrak{F}$ be a hereditary saturated formation and let $S$ be a $K$-$\mathfrak{F}$-subnormal subgroup of $G$. If $\mathbf{Z}_{\mathfrak{F}}(E) = 1$ for every subgroup $E$ of $G$ such that $S \leq E$ then $\mathbf{C}_{G}(D) \leq D$, where $D = S^{\mathfrak{F}}$ is the $\mathfrak{F}$-residual of $S$.

math.GR↗
arXiv2020

On $σ$-arithmetic graphs of finite groups

Let $G$ be a finite group and $σ$ a partition of the set of all? primes $\Bbb{P}$, that is, $σ=\{σ_i \mid i\in I \}$, where $\Bbb{P}=\bigcup_{i\in I} σ_i$ and $σ_i\cap σ_j= \emptyset $ for all $i\ne j$. If $n$ is an integer, we write $σ(n)=\{σ_i \mid σ_{i}\cap π(n)\ne \emptyset \}$ and $σ(G)=σ(|G|)$. We call a graph $Γ$ with the set of all vertices $V(Γ)=σ(G)$ ($G\ne 1$) a $σ$-arithmetic graph of $G$, and we associate with $G\ne 1$ the following three directed $σ$-arithmetic graphs: (1) the $σ$-Hawkes graph $Γ_{Hσ}(G)$ of $G$ is a $σ$-arithmetic graph of $G$ in which $(σ_i, σ_j)\in E(Γ_{Hσ}(G))$ if $σ_j\in σ(G/F_{\{σ_i\}}(G))$; (2) the $σ$-Hall graph $Γ_{σHal}(G)$ of $G$ in which $(σ_i, σ_j)\in E(Γ_{σHal}(G))$ if for some Hall $σ_i$-subgroup $H$ of $G$ we have $σ_j\in σ(N_{G}(H)/HC_{G}(H))$; (3) the $σ$-Vasil'ev-Murashko graph $Γ_{{\mathfrak{N}_σ}}(G)$ of $G$ in which $(σ_i, σ_j)\in E(Γ_{{\mathfrak{N}_σ}}(G))$ if for some ${\mathfrak{N}_{σ}}$-critical subgroup $H$ of $G$ we have $σ_i \in σ(H)$ and $σ_j\in σ(H/F_{\{σ_i\}}(H))$. In this paper, we study the structure of $G$ depending on the properties of these three graphs of $G$.

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arXiv2019

On a lattice characterization of finite soluble $PST$-groups

Let $\mathfrak{F}$ be a class of finite groups and $G$ a finite group. Let ${\cal L}_{\mathfrak{F}}(G)$ be the set of all subgroups $A$ of $G$ with $A^{G}/A_{G}\in \mathfrak{F}$. A chief factor $H/K$ of $G$ is $\mathfrak{F}$-central in $G$ if $(H/K)\rtimes (G/C_{G}(H/K)) \in\mathfrak{F}$. We study the structure of $G$ under the hypothesis that every chief factor of $G$ between $A_{G}$ and $A^{G}$ is $\mathfrak{F}$-central in $G$ for every subgroup $A\in {\cal L}_{\mathfrak{F}}(G)$. As an application, we prove that a finite soluble group $G$ is a $PST$-group if and only if $A^{G}/A_{G}\leq Z_{\infty}(G/A_{G})$ for every subgroup $A\in {\cal L}_{\mathfrak{N}}(G)$, where $\mathfrak{N}$ is the class of all nilpotent groups.

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arXiv2018

One application of the $σ$-local formations of finite groups

Throughout this paper, all groups are finite. Let $σ=\{σ_{i} | i\in I \}$ be some partition of the set of all primes $\Bbb{P}$. If $n$ is an integer, the symbol $σ(n)$ denotes the set $\{σ_{i} |σ_{i}\cap π(n)\ne \emptyset \}$. The integers $n$ and $m$ are called $σ$-coprime if $σ(n)\cap σ(m)=\emptyset$. Let $t > 1$ be a natural number and let $\mathfrak{F}$ be a class of groups. Then we say that $\mathfrak{F}$ is $Σ_{t}^σ$-closed provided $\mathfrak{F}$ contains each group $G$ with subgroups $A_{1}, \ldots , A_{t}\in \mathfrak{F}$ whose indices $|G:A_{1}|$, $\ldots$, $|G:A_{t}|$ are pairwise $σ$-coprime. In this paper, we study $Σ_{t}^σ$-closed classes of finite groups.

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arXiv2018

On $σ$-quasinormal subgroups of finite groups

Let $G$ be a finite group and $σ=\{σ_{i} | i\in I\}$ some partition of the set of all primes $\Bbb{P}$, that is, $σ=\{σ_{i} | i\in I \}$, where $\Bbb{P}=\bigcup_{i\in I} σ_{i}$ and $σ_{i}\cap σ_{j}= \emptyset $ for all $i\ne j$. We say that $G$ is $σ$-primary if $G$ is a $σ_{i}$-group for some $i$. A subgroup $A$ of $G$ is said to be: $σ$-subnormal in $G$ if there is a subgroup chain $A=A_{0} \leq A_{1} \leq \cdots \leq A_{n}=G$ such that either $A_{i-1}\trianglelefteq A_{i}$ or $A_{i}/(A_{i-1})_{A_{i}}$ is $σ$-primary for all $i=1, \ldots, n$, modular in $G$ if the following conditions hold: (i) $\langle X, A \cap Z \rangle=\langle X, A \rangle \cap Z$ for all $X \leq G, Z \leq G$ such that $X \leq Z$, and (ii) $\langle A, Y \cap Z \rangle=\langle A, Y \rangle \cap Z$ for all $Y \leq G, Z \leq G$ such that $A \leq Z$. In this paper, a subgroup $A$ of $G$ is called $σ$-quasinormal in $G$ if $L$ is modular and $σ$-subnormal in $G$. We study $σ$-quasinormal subgroups of $G$. In particular, we prove that if a subgroup $H$ of $G$ is $σ$-quasinormal in $G$, then for every chief factor $H/K$ of $G$ between $H^{G}$ and $H_{G}$ the semidirect product $(H/K)\rtimes (G/C_{G}(H/K))$ is $σ$-primary.

math.GR↗
arXiv2018

On one generalization of finite nilpotent groups

Let $σ=\{σ_{i} | i\in I\}$ be a partition of the set $\Bbb{P}$ of all primes and $G$ a finite group. A chief factor $H/K$ of $G$ is said to be $σ$-central if the semidirect product $(H/K)\rtimes (G/C_{G}(H/K))$ is a $σ_{i}$-group for some $i=i(H/K)$. $G$ is called $σ$-nilpotent if every chief factor of $G$ is $σ$-central. We say that $G$ is semi-$σ$-nilpotent (respectively weakly semi-$σ$-nilpotent) if the normalizer $N_{G}(A)$ of every non-normal (respectively every non-subnormal) $σ$-nilpotent subgroup $A$ of $G$ is $σ$-nilpotent. In this paper we determine the structure of finite semi-$σ$-nilpotent and weakly semi-$σ$-nilpotent groups.

math.GR↗
arXiv2017

On generalized $σ$-soluble groups

Let $σ=\{σ_{i} | i\in I\}$ be a partition of the set of all primes $\Bbb{P}$ and $G$ a finite group. Let $σ(G)=\{σ_{i} : σ_{i}\cap π(G)\ne \emptyset$. A set ${\cal H}$ of subgroups of $G$ is said to be a complete Hall $σ$-set of $G$ if every member $\ne 1$ of ${\cal H}$ is a Hall $σ_{i}$-subgroup of $G$ for some $i\in I$ and $\cal H$ contains exactly one Hall $σ_{i}$-subgroup of $G$ for every $i$ such that $σ_{i}\in σ(G)$. We say that $G$ is $σ$-full if $G$ possesses a complete Hall $σ$-set. A complete Hall $σ$-set $\cal H$ of $G$ is said to be a $σ$-basis of $G$ if every two subgroups $A, B \in\cal H$ are permutable, that is, $AB=BA$. In this paper, we study properties of finite groups having a $σ$-basis. In particular, we prove that if $G$ has a a $σ$-basis, then $G$ is generalized $σ$-soluble, that is, $G$ has a complete Hall $σ$-set and for every chief factor $H/K$ of $G$ we have $|σ(H/K)|\leq 2$. Moreover, answering to Problem 8.28 in [A.N. Skiba, On some results in the theory of finite partially soluble groups, Commun. Math. Stat., 4(3) (2016), 281--309], we prove the following Theorem A. Suppose that $G$ is $σ$-full. Then every complete Hall $σ$-set of $G$ forms a $σ$-basis of $G$ if and only if $G$ is generalized $σ$-soluble and for the automorphism group $G/C_{G}(H/K)$, induced by $G$ on any its chief factor $H/K$, we have either $σ(H/K)=σ(G/C_{G}(H/K))$ or $σ(H/K) =\{σ_{i}\}$ and $G/C_{G}(H/K)$ is a $σ_{i} \cup σ_{j}$-group for some $i\ne j$.

math.GR↗
arXiv2017

A Robinson characterization of finite $PσT$-groups

Let $σ=\{σ_{i} | i\in I\}$ be some partition of the set of all primes $\Bbb{P}$ and let $G$ be a finite group. Then $G$ is said to be $σ$-full if $G$ has a Hall $σ_{i}$-subgroup for all $i$. A subgroup $A$ of $G$ is said to be $σ$-permutable in $G$ provided $G$ is $σ$-full and $A$ permutes with all Hall $σ_{i}$-subgroups $H$ of $G$ (that is, $AH=HA$) for all $i$. We obtain a characterization of finite groups $G$ in which $σ$-permutability is a transitive relation in $G$, that is, if $K$ is a $σ$-permutable subgroup of $H$ and $H$ is a $σ$-permutable subgroup of $G$, then $K$ is a $σ$-permutable subgroup of $G$.

math.GR↗
arXiv2017

Finite groups with systems of $K$-$\frak{F}$-subnormal subgroups

Let $\frak {F}$ be a class of group. A subgroup $A$ of a finite group $G$ is said to be $K$-$\mathfrak{F}$-subnormal in $G$ if there is a subgroup chain $$A=A_{0} \leq A_{1} \leq \cdots \leq A_{n}=G$$ such that either $A_{i-1} \trianglelefteq A_{i}$ or $A_{i}/(A_{i-1})_{A_{i}} \in \mathfrak{F}$ for all $i=1, \ldots , n$. A formation $\frak {F}$ is said to be $K$-lattice provided in every finite group $G$ the set of all its $K$-$\mathfrak{F}$-subnormal subgroups forms a sublattice of the lattice of all subgroups of $G$. In this paper we consider some new applications of the theory of $K$-lattice formations. In particular, we prove the following Theorem A. Let $\mathfrak{F}$ be a hereditary $K$-lattice saturated formation containing all nilpotent groups. (i) If every $\mathfrak{F}$-critical subgroup $H$ of $G$ is $K$-$\mathfrak{F}$-subnormal in $G$ with $H/F(H)\in {\mathfrak{F}}$, then $G/F(G)\in {\mathfrak{F}}$. (ii) If every Schmidt subgroup of $G$ is $K$-$\mathfrak{F}$-subnormal in $G$, then $G/G_{\mathfrak{F}}$ is abelian.

math.GR↗
arXiv2017

On the lattice of the $σ$-permutable subgroups of a finite group

Let $σ=\{σ_{i} | i\in I\}$ be some partition of the set of all primes $\Bbb{P}$, $G$ a finite group and $σ(G) =\{σ_{i} |σ_{i}\cap π(G)\ne \emptyset \}$. A set ${\cal H}$ of subgroups of $G$ is said to be a complete Hall $σ$-set of $G$ if every member $\ne 1$ of ${\cal H}$ is a Hall $σ_{i}$-subgroup of $G$ for some $σ_{i}\in σ$ and ${\cal H}$ contains exactly one Hall $σ_{i}$-subgroup of $G$ for every $σ_{i}\in σ(G)$. A subgroup $A$ of $G$ is said to be $σ$-permutable in $G$ if $G$ possesses a complete Hall $σ$-set and $A$ permutes with each Hall $σ_{i}$-subgroup $H$ of $G$, that is, $AH=HA$ for all $i \in I$. We characterize finite groups with distributive lattice of the $σ$-permutable subgroups.

math.GR↗
arXiv2017

Characterizations of some classes of finite $σ$-soluble $PσT$-groups

Let $σ=\{σ_{i} | i\in I\}$ be some partition of the set of all primes $\Bbb{P}$ and $G$ a finite group. $G$ is said to be $σ$-soluble if every chief factor $H/K$ of $G$ is a $σ_{i}$-group for some $i=i(H/K)$. A set ${\cal H}$ of subgroups of $G$ is said to be a complete Hall $σ$-set of $G$ if every member $\ne 1$ of ${\cal H}$ is a Hall $σ_{i}$-subgroup of $G$ for some $σ_{i}\in σ$ and ${\cal H}$ contains exactly one Hall $σ_{i}$-subgroup of $G$ for every $i \in I$ such that $σ_{i}\cap π(G)\ne \emptyset$. A subgroup $A$ of $G$ is said to be $σ$-permutable in $G$ if $G$ has a complete Hall $σ$-set $\cal H$ such that $AH^{x}=H^{x}A$ for all $x\in G$ and all $H\in \cal H$. We obtain characterizations of finite $σ$-soluble groups $G$ in which $σ$-permutability is a transitive relation in $G$.

math.GR↗
arXiv2017

On $H_σ$-permutably embedded and $H_σ$-subnormaly embedded subgroups of finite groups

Let $G$ be a finite group. Let $σ=\{σ_{i} | i\in I\}$ be a partition of the set of all primes $\Bbb{P}$ and $n$ an integer. We write $σ(n) =\{σ_{i} |σ_{i}\cap π(n)\ne \emptyset \}$, $σ(G) =σ(|G|)$. A set $ {\cal H}$ of subgroups of $G$ is said to be a complete Hall $σ$-set of $G$ if every member of ${\cal H}\setminus \{1\}$ is a Hall $σ_{i}$-subgroup of $G$ for some $σ_{i}$ and ${\cal H}$ contains exact one Hall $σ_{i}$-subgroup of $G$ for every $σ_{i}\in σ(G)$. A subgroup $A$ of $G$ is called: (i) a $σ$-Hall subgroup of $G$ if $σ(|A|) \cap σ(|G:A|)=\emptyset$; (ii) $σ$-permutable in $G$ if $G$ possesses a complete Hall $σ$-set ${\cal H}$ such that $AH^{x}=H^{x}A$ for all $H\in {\cal H}$ and all $x\in G$. We say that a subgroup $A$ of $G$ is $H_σ$-permutably embedded in $G$ if $A$ is a $σ$-Hall subgroup of some $σ$-permutable subgroup of $G$. We study finite groups $G$ having an $H_σ$-permutably embedded subgroup of order $|A|$ for each subgroup $A$ of $G$. Some known results are generalized.

math.GR↗
arXiv2016

On finite $PσT$-groups

Let $σ=\{σ_{i} | i\in I\}$ be some partition of the set of all primes $\Bbb{P}$ and $G$ a finite group. $G$ is said to be \emph{$σ$-soluble} if every chief factor $H/K$ of $G$ is a $σ_{i}$-group for some $i=i(H/K)$. A set ${\cal H}$ of subgroups of $G$ is said to be a \emph{complete Hall $σ$-set} of $G$ if every member $\ne 1$ of ${\cal H}$ is a Hall $σ_{i}$-subgroup of $G$ for some $σ_{i}\in σ$ and ${\cal H}$ contains exact one Hall $σ_{i}$-subgroup of $G$ for every $i \in I$ such that $σ_{i}\cap π(G)\ne \emptyset$. A subgroup $A$ of $G$ is said to be \emph{$σ$-permutable} or \emph{$σ$-quasinormal} in $G$ if $G$ has a complete Hall $σ$-set $\cal H$ such that $AH^{x}=H^{x}A$ for all $x\in G$ and all $H\in \cal H$. We obtain a characterization of finite $σ$-soluble groups $G$ in which $σ$-quasinormality is a transitive relation in $G$.

math.GR↗
arXiv2016

On $σ$-semipermutable subgroups of finite groups

Let $σ=\{σ_{i} | i\in I\}$ be some partition of the set of all primes $\Bbb{P}$, $G$ a finite group and $σ(G) =\{σ_{i} |σ_{i}\cap π(G)\ne \emptyset \}$. A set ${\cal H}$ of subgroups of $G$ is said to be a \emph{complete Hall $σ$-set} of $G$ if every member $\ne 1$ of ${\cal H}$ is a Hall $σ_{i}$-subgroup of $G$ for some $σ_{i}\in σ$ and ${\cal H}$ contains exact one Hall $σ_{i}$-subgroup of $G$ for every $σ_{i}\in σ(G)$. A subgroup $H$ of $G$ is said to be: \emph{$σ$-semipermutable in $G$ with respect to ${\cal H}$} if $HH_{i}^{x}=H_{i}^{x}H$ for all $x\in G$ and all $H_i\in {\cal H}$ such that $(|H|, |H_{i}|)=1$; \emph{$σ$-semipermutable in $G$} if $H$ is $σ$-semipermutable in $G$ with respect to some complete Hall $σ$-set of $G$. We study the structure of $G$ being based on the assumption that some subgroups of $G$ are $σ$-semipermutable in $G$.

math.GR↗
arXiv2016

Finite groups whose $n$-maximal subgroups are $σ$-subnormal

Let $σ=\{σ_{i} | i\in I\}$ be some partition of the set of all primes $\Bbb{P}$. A set ${\cal H}$ of subgroups of $G$ is said to be a \emph{complete Hall $σ$-set} of $G$ if every member $\ne 1$ of ${\cal H}$ is a Hall $σ_{i}$-subgroup of $G$, for some $i\in I$, and $\cal H$ contains exact one Hall $σ_{i}$-subgroup of $G$ for every $σ_{i}\in σ(G)$. A subgroup $H$ of $G$ is said to be: \emph{$σ$-permutable} or \emph{$σ$-quasinormal} in $G$ if $G$ possesses a complete Hall $σ$-set set ${\cal H}$ such that $HA^{x}=A^{x}H$ for all $A\in {\cal H}$ and $x\in G$: \emph{$σ$-subnormal} in $G$ if there is a subgroup chain $A=A_{0} \leq A_{1} \leq \cdots \leq A_{t}=G$ such that either $A_{i-1}\trianglelefteq A_{i}$ or $A_{i}/(A_{i-1})_{A_{i}}$ is a finite $σ_{i}$-group for some $σ_{i}\in σ$ for all $i=1, \ldots t$. If each $n$-maximal subgroup of $G$ is $σ$-subnormal ($σ$-quasinormal, respectively) in $G$ but, in the case $ n > 1$, some $(n-1)$-maximal subgroup is not $σ$-subnormal (not $σ$-quasinormal, respectively)) in $G$, we write $m_σ(G)=n$ ($m_{σq}(G)=n$, respectively). In this paper, we show that the parameters $m_σ(G)$ and $m_{σq}(G)$ make possible to bound the $σ$-nilpotent length $ \ l_σ(G)$ (see below the definitions of the terms employed), the rank $r(G)$ and the number $|π(G)|$ of all distinct primes dividing the order $|G|$ of a finite soluble group $G$. We also give conditions under which a finite group is $σ$-soluble or $σ$-nilpotent, and describe the structure of a finite soluble group $G$ in the case when $m_σ(G)=|π(G)|$. Some known results are generalized.

math.GR↗