Frattini supplements and Frat- series

‎In this study‎, ‎Frattini supplement subgroup and Frattini supplemented group‎ ‎are defined by Frattini subgroup‎. ‎By these definitions‎, ‎it's shown that‎ ‎finite abelian groups are Frattini supplemented and every conjugate of a‎ ‎Frattini supplement of a subgroup is also a Frattini supplement‎. ‎A group action‎ ‎of a group is defined 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 defined 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.