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Inductivity of the lattice of $\sigma$-local fitting classes
N. N. Vorob'ev,
I. I. Staselka Vitebsk State University named after P. M. Masherov
Abstract:
All groups under consideration are finite. Let
$\sigma = \{\sigma_i \mid i \in I\}$ be a partition of the set
$\mathbb{P}$ of all primes, and let
$f$ be any function from
$\sigma$ to Fitting classes; such a function is called a Hartley
$\sigma$-function (or, briefly, an
$H_\sigma$-function). Consider the class
$$ LR_{\sigma}(f)=\bigl(G \mid G=1 \text{ or } G \ne 1 \text{ and } G^{\mathfrak{G}_{\sigma_i}\mathfrak{G}_{\sigma_i'}} \in f(\sigma_i) \text{ for all } \sigma_i \in \sigma(G)\bigr) $$
of groups. If a Fitting class
$\mathfrak{F}$ is such that
$\mathfrak{F}=LR_{\sigma}(f)$ for some
$H_\sigma$-function
$f$, then
$\mathfrak{F}$ is called a
$\sigma$-local Fitting class and
$f$, a
$\sigma$-local definition of
$\mathfrak{F}$. Given a complete lattice
$\Theta$ of Fitting classes, the least upper bound of any set
$\{\mathfrak{F}_j \mid j \in J\}$ of elements of
$\Theta^{\sigma_l}$ is denoted by
$$ \bigvee_{\Theta^{\sigma_l}}(\mathfrak{F}_j \mid j \in J). $$
The lattice
$\Theta^{\sigma_l}$ is said to be inductive if, given any set
$\{\mathfrak{F}_j=LR_\sigma(f_j) \mid j \in J\}$ of Fitting classes
$\mathfrak{F}_j \in \Theta^{\sigma_l}$ and any set
$\{f_j \mid j \in J\}$ of
$\Theta$-valued
$H_\sigma$-functions
$f_j$, where each
$f_j$ is an integrated
$H_\sigma$-function of the Fitting class
$\mathfrak{F}_j$, the relation
$$ \bigvee_{\Theta^{\sigma_l}}(\mathfrak{F}_j \mid j \in J) =LR_\sigma\bigl(\bigvee_\Theta(f_j \mid j \in J)\bigr) $$
holds, where
$$ \bigvee_\Theta(f_j \mid j \in J) $$
denotes the
$H_\sigma$-function
$f$ such that
$f(\sigma_i)$ is the least upper bound of
$\{f_j(\sigma_i) \mid j \in J\}$ in
$\Theta$ if
$$ \bigcup_{j \in J}f_j(\sigma_i) \ne \varnothing $$
and
$f(\sigma_i)=\varnothing$ otherwise. It is proved that the lattice of all
$\sigma$-local Fitting classes is inductive.
Keywords:
finite group, Fitting class, complete lattice of Fitting classes,
Hartley
$\sigma$-function,
$\sigma$-local Fitting class, inductive lattice of
Fitting classes.
UDC:
512.542
MSC: 20F17,
20D10,
06B23 Received: 15.10.2023
Revised: 12.01.2025
DOI:
10.4213/mzm14174
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