Finite Groups with Seminormal Sylow Subgroups

In this paper, we prove the following theorem: Let p be a prime number, P a Sylow p-subgroup of a group G and pi = pi(G) \ {p}. If P is seminormal in G, then the following statements hold: 1) G is a p-soluble group and P' <= O-p(G); 2) l(p)(G) <= 2 and l(pi)(G) <= 2; 3) if a pi-Hall subgroup of G is q-supersoluble for some q is an element of pi, then G is q-supersoluble.