Conditions on p-subgroups implying p-nilpotence or p-supersolvability

Abstract

We discuss a condition on a p-subgroup H of a finite group G that is both more general and easier to work with than the assumption that H is weakly s-permutable in G. Our condition is that ${U \cap H \triangleleft U}$, where U =  O ^p(H), and we assume that this condition holds for every subgroup H of order d that is normal in some fixed Sylow p-subgroup P of G, where d >  1 is a fixed power of p dividing |G|. We show in this situation that either G is p-supersolvable or else ${|P \cap U| > d}$, and we derive some corollaries that extend known results concerning weakly s-permutable subgroups.