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.
