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MATHEMATICS
On one generalization of $\sigma$-local and Baer-local formations
V. G. Safonova,
I. N. Safonovaa,
A. N. Skibab a Belarusian State University, Minsk
b F. Scorina Gomel State University
Abstract:
Throughout this paper, all groups are finite and
$G$ is a group. Let
$\sigma=\{\sigma_i\mid i\in I\}$ be some partition of the set of all primes
$\mathbb{P}$.
Then $\sigma(G)=\{\sigma_i\mid \sigma_i\cap\pi(G)\ne\varnothing\}$;
$\sigma^+(G)=\{\sigma_i\mid G \text{ has a chief factor } H/K, \text{ such that } \sigma(H/K)=\{\sigma_i\}\}$. The group
$G$ is said
to be:
$\sigma$-
primary if
$G$ is
$\sigma_i$-group for some
$i$;
$\sigma$-
soluble if every chief factor of
$G$ is
$\sigma$-primary. The symbol
$R_\sigma(G)$ denotes the product of all normal
$\sigma$-soluble subgroups of
$G$. The chief factor
$H/K$ of
$G$ is said to be:
$\sigma$-
central (in
$G$) if
$(H/K)\rtimes(G/C_G(H/K))$ is
$\sigma$-primary; a
$\sigma_i$-
factor if
$H/K$ is a
$\sigma_i$-group. We say that
$G$ is:
$\sigma$-
nilpotent if every chief factor of
$G$ is
$\sigma$-central;
generalized $\{\sigma_i\}$-
nilpotent if every chief
$\sigma_i$-factor of
$G$ is
$\sigma$-central. The symbol
$F_{\{g\sigma_i\}}(G)$ denotes the product of all normal generalized
$\{\sigma_i\}$-nilpotent subgroups of
$G$. We call any function
$f$ of the form
$f:\sigma\cup\{\varnothing\}\to\{\text{formations of groups}\}$, where
$f(\varnothing)\ne\varnothing$,
a generalized formation $\sigma$-
function and we put
$$
BLF_\sigma(f)=(G\mid G/R_\sigma(G)\in f(\varnothing) \text{ and } G/F_{\{g\sigma_i\}}(G)\in f(\sigma_i) \text{ for all }\sigma_i\in\sigma^+(G)).
$$
If for some generalized formation
$\sigma$-function
$f$ we have
$\mathfrak{F}=BLF_\sigma(f)$, then we say that the class
$\mathfrak{F}$ is
Baer-
$\sigma$-
local and
$f$ is a
generalized $\sigma$-
local definition of $\mathfrak{F}$. In this paper, we describe basic properties, examples, and some applications of Baer-
$\sigma$-local formations.
Keywords:
finite group, generalized formation $\sigma$-function, Baer-$\sigma$-local formation, generalized $\{\sigma_i\}$-nilpotent group, Gaschütz product.
UDC:
512.542 Received: 01.11.2019
Language: English
Fulltext:
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