Finite groups with formational subnormal primary subgroups of bounded exponent (Q6587412)

Let \(\mathfrak U_k\) be the class of all finite supersoluble groups whose exponent is not divisible by \(p^{k+1}\) for any prime \(p.\) The authors investigate the classes \(\mathrm{w}\mathfrak U_k\) and \(\mathrm{v}\mathfrak U_k\) consisting of all finite groups in which every Sylow and every cyclic primary subgroup is \(\mathfrak U_k\)-subnormal. They prove that although the formation \(\mathfrak U_k\) is not saturated, \(\mathrm{w}\mathfrak U_k\) and \(\mathrm{v}\mathfrak U_k\) are subgroup-closed saturated formations. They also obtain some characterizations of these formations: for example \(G \in \mathrm{w}\mathfrak U_k\) if and only if \(A/\Phi(A) \in \mathfrak U_k\) for every metanilpotent subgroup \(A\) of \(G\), while \(G \in \mathrm{v}\mathfrak U_k\) if and only if \(B/\Phi(B) \in \mathfrak U_k\) for every subgroup \(B\) of \(G\) whose derived subgroup is nilpotent.