(1) Department of Mathematics, Shanxi University, 030006 Taiyuan, China;(2) Department of Mathematics, The Chinese University of Hong Kong, Shatin, N. T., Hong Kong, China (SAR)
Abstract:
We call a subgroup H of a finite group G c-supplemented in G if there exists a subgroup K of G such that G = HK and H ∩ K ⩽ core(H). In this paper it is proved that a finite group G is p-nilpotent if G is S4-free and every minimal subgroup of P ∩ GN is c-supplemented in NG(P), and when p = 2 P is quaternion-free, where p is the smallest prime number dividing the order of G, P a Sylow p-subgroup of G. As some applications of this result, some known results are generalized.