Frattini supplements and Frat- series

In this studyý, ýFrattini supplement subgroup and Frattini supplemented groupý ýare defined by Frattini subgroupý. ýBy these definitionsý, ýit's shown thatý ýfinite abelian groups are Frattini supplemented and every conjugate of aý ýFrattini supplement of a subgroup is also a Frattini supplementý. ýA group actioný ýof a group is defined over the set of Frattini supplements of a normalý ýsubgroup of the group by conjugation and in this study new characterizationý ýof primitivity of groups has obtained in terms of Frattini supplementedý ýgroups by this actioný. ýMoreover, Frat-series of a group is defined based oný ýFrattini supplements of normal subgroups of the group and it is shown thatý ýsubgroups and factor groups of groups with Frat-series also have Frat-seriesý ýunder some special conditionsý. ýFurthermoreý, ýwe determined a characterization of soluble groups which have Frat-series.