p-子群皆p-拟正规或自正规的有限群

FINITE GROUPS WHOSE p SUBGROUPS ARE EITHER p OUASI NORMAL OR SELFNORMAL

本文首先证明了ｐ－子群皆ｐ－拟正规或自正规的有限群的分类定理．由此，得到了每个子群皆Ｓ－拟正规或自正规的有限群的分类定理．

This paper proves the following theorem: For finite group G, the following three statements are equivalent: (1) for each prime p∈π(G), every p subgroup of G is p quasi normal or self normal in G;(2) for each prime p∈π(G), ever Sylow p subgroup of G and all its maximal subgroups are p quasi normal or selfnormal in G; (3) G is one of the following two classes of groups: Ⅰ. nilpotent groups; Ⅱ. G=QH, in which, Q=〈x〉∈Syl q(G), H is the normal abelian q com plement of G, and 〈x q〉=O q(G)=Z(G), x  induces a fixed point free automorphism of H by conjugating. From this theroem, we obtain the classification theorem of the finite groups with only S quasi normal or self normal subgroups. The main theorems in the papers of Frattahi (1974) and Zhang Wang (1995) can be obtained by our results.