On $\mathrm K$-$\mathbb P$-Subnormal Subgroups of Finite Groups

A subgroup $H$ of a group $G$ is said to be $\mathrm K$-$\mathbb P$-subnormal in $G$ if $H$ can be joined to the group by a chain of subgroups each of which is either normal in the next subgroup or of prime index in it. Properties of $\mathrm K$-$\mathbb P$-subnormal subgroups are obtained. A class of finite groups whose Sylow $p$-subgroups are $\mathrm K$-$\mathbb P$-subnormal in $G$ for every $p$ in a given set of primes is studied. Some products of $\mathrm K$-$\mathbb P$-subnormal subgroups are investigated.