Frattini supplements and Frat series

In this study, Frattini supplement subgroup and Frattini sup- plemented group are de ned by Frattini subgroup. By these de nitions, it s shown that nite abelian groups are Frattini supplemented and ev- ery conjugate of a Frattini supplement of a subgroup is also a Frattini supplement. A group action of a group is de ned over the set of Frattini supplements of a normal subgroup of the group by conjugation and in this study new characterization of primitivity of groups has obtained in terms of Frattini supplemented groups by this action. Moreover, Frat-series of a group is de ned based on Frattini supplements of normal subgroups of the group and it is shown that subgroups and factor groups of groups with Frat-series also have Frat-series under some special conditions. Furthermore, we determined a characterization of soluble groups which have Frat-series.