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

Abstract

Let \(G\) be a finite group and \(\sigma=\{\sigma_i | i\in I\}\) be some partition of the set of all primes. A subgroup \(A\) of \(G\) is said to \(K\)-\(\frak{N}_{\sigma}\)-subnormal in \(G\) if there is a subgroup chain \(A=A_o\leq A_1 \leq \cdots \leq A_n=G\) such that either \(A_{i-1} \unlhd A_i\) or \(A_i/ (A_{i-1})_{A_i}\in \frak{N}_{\sigma}\) for all \(i=1,\ldots, n\), where \(\frak{N}_{\sigma}\) is a hereditary \(K\)-lattice saturated formation of all \(\sigma\)-nilpotent groups. The formation \(\frak{N}_{\sigma}\) is called \(K\)-lattice if in every finite group \(G\) the set \(\mathcal{L}_{K\frak{N}_{\sigma}}(G)\), of all \(K\)-\(\frak{N}_{\sigma}\)-subnormal subgroup of \(G\), is a sublattice of the lattice \(\mathcal{L}(G)\) of all subgroups of \(G\). In this paper we prove that if every Schmidt subgroup of \(G\) is \(K\)-\(\frak{N}_{\sigma}\)-subnormal subgroup of \(G\), then the commutator subgroup \(G'\) of \(G\) belongs to hereditary \(K\)-lattice saturated formation \(\frak{N}_{\sigma}\).