On finite groups with \(K\)-\(\mathfrak{N}_{\sigma}\)-subnormal Schmidt subgroups (Q6585986)

Let \(G\) be a finite group, \(\mathbb{P}\) the set of all primes and let \(\sigma=\{\sigma_{i} \mid i \in I \}\) be a partition of \(\mathbb{P}\). A subgroup \(A\) is said to \(K\)-\(\mathfrak{N}_{\sigma}\)-subnormal in \(G\) if there is a subgroup chain \(A=A_{0} \leq A_{1} \leq \dots \leq A_{n}=G\) such that either \(A_{i-1} \trianglelefteq A_{i}\) or \(A_{i}/(A_{i-1})_{A_{i}} \in \mathfrak{N}_{\sigma}\) for all \(i \in \{1, \ldots, n\}\) where \(\mathfrak{N}_{\sigma}\) is a hereditary \(K\)-lattice saturated formation of all \(\sigma\)-nilpotent groups. The formation \(\mathfrak{N}_{\sigma}\) is called \(K\)-lattice if in every finite group \(G\) the set \(\mathcal{L}_{K\mathfrak{N}_{\sigma}}(G)\) of all \(K\)-\(\mathfrak{N}_{\sigma}\)-subnormal subgroups of \(G\) is a sublattice of the lattice \(\mathcal{L}(G)\) of all subgroups of \(G\).\N\NA non-nilpotent finite group whose all proper subgroups are nilpotent is called a Schmidt group. The main result in the paper under review is the following. If every Schmidt subgroup of \(G\) is \(K\)-\(\mathfrak{N}_{\sigma}\)-subnormal in \(G\), then the commutator subgroup \(G'\) belongs to hereditary \(K\)-lattice saturated formation \(\mathfrak{N}_{\sigma}\).