Abstract
Let \(t\) be a chosen positive integer. A subgroup \(H\) of a finite group \(G\) is said to be \(\mathrm{K}\) - \(\mathbb{P}_{t}\) -subnormal in \(G\) if there exists a chain of subgroups \(H=H_{ 0} \leq H_{1} \leq \cdots \leq H_{m-1} \leq H_{m}=G\) such that either \(H_{i-1}\) is normal in \(H_{i}\) or \(|H_{i} : H_{i-1}|\) is some prime \(p\) and \(p-1\) is not divisible by \((t+1)\) st powers of primes for any \(i=1,\dots, m\) . In this paper, properties of \(\mathrm{K}\) - \(\mathbb{P}_{t}\) -subnormal subgroups are obtained. It is proved that all (supersolvable) groups with \(\mathrm{K}\) - \(\mathbb{P}_{t}\) -subnormal Sylow subgroups form a hereditary saturated formation, and its local definition is found.
