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BULLETIN OF THE IRANIAN MATHEMATICAL SOCIETY, vol.43, no.3, pp.747-753, 2017 (SCI-Expanded)
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.