A Characterisation of Powerfully Embedded Normal Subgroups of ap-Group Original Research Article

Letpbe a prime number andGbe ap-group.[6]have introduced the notion of a powerfully embedded subgroup ofG. Our main result gives a characterisation of those powerfully embedded subgroups which are contained in the Frattini subgroup ofG, whenpis odd. (Whenp = 2, the subgroups characterised have a weaker property than powerful embedding.) This result also yields information on the structure of “d-maximal” 2-groups.