Finite groups in which generalized normality is a transitive relation
In this paper, we discuss some well-known results and some open problems of the theory of $σ$-properties of a group related to the study of generalized $T$-groups.
math.GR↗
arXiv subjects
Inna N. Safonova
Publications and source records attributed to Inna N. Safonova.
On $τ$-closed $n$-multiply $σ$-local formations of finite groups
All groups under consideration are finite. Let $σ=\{σ_i \mid i\in I \}$ be some partition of the set of $\mathbb{P}$, $G$ be a group, and $\mathfrak F$ be a class of groups. Then $σ(G)=\{σ_i\mid σ_i\cap π(G)\ne \emptyset\} $ and $σ(\mathfrak F)=\cup_{G\in \mathfrak F}σ(G).$ A function $f$ of the form $f:σ\to\{\text{formations of groups}\}$ is called a formation $σ$-function. For any formation $σ$-function $f$ the class $LF_σ(f)$ is defined as follows: $$ LF_σ(f)=(G \text{ is a group } \mid G=1 \text{ or } G\ne 1\ \text{ and }\ G/O_{σ_i', σ_i}(G) \in f(σ_i) \text{ for all } σ_i \in σ(G)). $$ If for some formation $σ$-function $f$ we have $\mathfrak F=LF_σ(f),$ then $\mathfrak F$ is called $σ$-local, $f$ is called a $σ$-local definition of $\mathfrak F.$ Every formation is called 0-multiply $σ$-local. For $n > 0,$ a formation $\mathfrak F$ is called $n$-multiply $σ$-local provided either $\mathfrak F=(1)$ or $\mathfrak F=LF_σ(f),$ where $f(σ_i)$ is $(n-1)$-multiply $σ$-local for all $σ_i\in σ(\mathfrak F).$ Let $τ(G)$ be a set of subgroups of $G$ such that $G\in τ(G)$. Then $τ$ is called a subgroup functor if for every epimorphism $φ$ : $A \to~B$ and any groups $H\inτ(A)$ and $T\inτ(B)$ we have $H^φ\inτ(B)$ and $T^{φ^{-1}}\inτ(A)$. A class $\mathfrak F$ is called $τ$-closed if $τ(G)\subseteq\mathfrak F$ for all $G\in\mathfrak F$. We describe some properties of $τ$-closed $n$-multiply $σ$-local formations, as well as we prove that the set $l^τ_{σ_n}$ of all $τ$-closed $n$-multiply $σ$-local formations forms a complete modular algebraic lattice. In addition, we proof that $l^τ_{σ_n}$ is $σ$-inductive and $\mathfrak G$-separable.
math.GR↗
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.
math.GR↗