Abstract:

A subgroup H of a finite group G is said to be normally embedded in G if, for every prime p dividing the order of H, there exists a normal subgroup K of G such that a Sylow p- subgroup of H is also a Sylow p- subgroup of K。 This paper assumes that every non-cyclic Sylow subgroup P of G has a subgroup D such that 1<|D|<|P| and all subgroups H of P with order |H|=|D| and with 2|D|(if P is a non-abelian 2-group and |P:D|>2) are normally embedded in G, and some sufficient conditions are obtained on G to be p-nilpotent groups and supersolvable groups. Moreover, some of them are extended to formations.

Key words: normally embedded subgroups; p- normally embedded subgroups; p-nilpotent groups; supersolvable groups