Inductivity of the Lattice of $$\sigma$$-Local Fitting Classes
Nikolay Nikolayevich Vorob'ev, I. I. Staselka · Mathematical Notes · 2025
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 $$LR_{\sigma}(f)=\bigl(G \mid G=1 \text{ or } G e 1 \text{ and } G^{\mathfrak{G}_{\sigma_i}\mathfrak{G}_{\sigma_i'}} \in f(\sigma_i) \text{ for all } \sigma_i \in \sigma(G)\bigr)$$ 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 $$\bigcup_{j \in J}f_j(\sigma_i) e \varnothing$$ and $$f(\sigma_i)=\varnothing$$ otherwise. It is proved that the lattice of all $$\sigma$$ -local Fitting classes is inductive.