Abnormality Criteria for p-Complements

Abstract

It is proved that for any finite group G possessing a p-complement H for some prime number p, the following assertions are equivalent: (1) all p-complements of G are self-normalizable; (2) all p-complements of G are abnormal; (3) the subgroup H is abnormal in G; (4) N_G(HX) = HX for any $X\underset{―}{⊲}G$; (5) G does not contain central chief p-factors.