Frattini supplements and Frat- series

Document Type : Research Paper


Faculty of Arts and Sciences‎, ‎Department of Mathematics‎, ‎Ondokuz Mayis University‎, ‎Samsun‎, ‎Turkey.


‎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.


Main Subjects

D.J.S. Robinson, A Course in the Theory of Groups, Springer-Verlag, New York, 1982.
J.D. Dixon and B. Mortimer, Permutation Groups, Springer-Verlag, New York, 1996.
J. Hausen, Supplemented nilpotent groups, Rend. Sem. Mat. Univ. Padova  65 (1981), no. 3, 35--46.