Finite Groups with DOUBLE-STRUCK CAPITAL P-Subnormal Sylow Subgroups

Let DOUBLE-STRUCK CAPITAL P be the set of all prime numbers. A subgroup H of a finite group G is called DOUBLE-STRUCK CAPITAL P-subnormal if either H = G or there exists a chain of subgroups H = H-0 <= H-1 <= horizontal ellipsis <= H-n = G such that |H-i : Hi - 1| is an element of DOUBLE-STRUCK CAPITAL P, 1 <= i <= n. We prove that any finite group with DOUBLE-STRUCK CAPITAL P-subnormal Sylow p-subgroup of odd order is p-solvable and any group with DOUBLE-STRUCK CAPITAL P-subnormal generalized Schmidt subgroups is metanilpotent.