Finite groups with seminormal Sylow subgroups

Abstract

In this paper, we prove the following theorem: Let p be a prime number, P a Sylow p-subgroup of a group G and π = π(G) \ {p}. If P is seminormal in G, then the following statements hold: 1) G is a p-soluble group and P′ ≤ Op(G); 2) lp(G) ≤ 2 and lπ(G) ≤ 2; 3) if a π-Hall subgroup of G is q-supersoluble for some q ∈ π, then G is q-supersoluble.