On properties of formations of quasinilpotent groups
A finite group \(G\) is said to be quasinilpotent if \(C_ G(H/K)H = G\), for any chief factor \(H/K\) of \(G\). Let \({\mathcal N}^*\) be the class of all quasinilpotent finite groups and \(\mathcal N\) the class of all nilpotent finite groups. Both \(\mathcal N\) and \({\mathcal N}^*\) are radical formations. The paper confirms the dualism between \(\mathcal N\) and \({\mathcal N}^*\). The author proves that: i) for any two subnormal subgroups \(A\) and \(B\) of any arbitrary finite group \(G\), we have \(\langle A, B\rangle^{{\mathcal N}^*} \equiv \langle A^{{\mathcal N}^*}, B^{{\mathcal N}^*}\rangle\); ii) the quasinilpotent radical of a subnormal subgroup of an arbitrary finite group \(G\) commutes with any of its other subnormal subgroups.
scientific article; zbMATH DE number 567874
scientific article