Generalized Frattini Properties And Related Problems

The Frattini subgroup and Frattini properties are the important research subjects in group theory.For any group G , the Frattini subgroup Frat (G ) is the intersection of all the maximal subgroups of G . Letχbe a class of groups. If for anyχ-group G G Frat (G ) has P→G has P , then P is called a Frattini property ofχ-groups.There are three research topics in this paper. Firstly, we investigate the generalized Frattini subgroup of amalgamated free products of groups and generalize a result of M.K.Azarian [1] on the lower near Frattini subgroup of free products of groups with cyclic amalgamations to the upper near Frattini subgroup of the same type of groups. This gives a positive answer to Open Question 1 and a partial answer to Open Question 2 in [1]. Also, we consider the f Frattini subgroup of free products of groups with cyclic amalgamations and obtain a similar result. Secondly, we introduce the definition of the f Frattini property and prove that local solubility, local nilpotency and local supersolubility are all f Frattini properties of FC-groups. Finally, we study the relationship between the nearly splitting property of some type of normal subgroups and the generalized Frattini subgroup. And we extend two results of K.Muthuvel [11] on the lower near Frattini subgroup to the upper near Frattini subgroup.