In this paper we continue our study of the Frattini p-subalgebra of a Lie />-algebra L. We show first that if L is solvable then its Frattini p-subalgebra is an ideal of L. We then consider Lie p-algebras L in which X. is nilpotent and find necessary and sufficient conditions for the Frattini p-subalgebra to be trivial. From this we deduce, in particular, that in such an algebra every ideal als...

1. The universal p-Frattini cover. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2. The p-Frattini module. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3. Restriction to the normalizer of a p-Sylow. . . . . . . . . . . . . . . . . . . . . . . 8 4. Asymptotics of the p-Frattini modules Mn . . . . . . . . . . . . . . . . . . . . ....

‎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 ‎subgro...

An extension of a group A by a group G is thought of here simply as a group H containing A as a normal subgroup with quotient H/A isomorphic to G. It is called a central Frattini extension if A is contained in the intersection of the centre and the Frattini subgroup of H . The result of the paper is that, given a finite abelian A and finite G, there exists a central Frattini extension of A by G...

We give a complete characterization of finitely generated primitive groups in several settings including linear groups, subgroups of mapping class groups, groups acting minimally on trees and convergence groups. The latter category includes as a special case Kleinian groups as well as finitely generated subgroups of word hyperbolic groups. As an application we calculate the Frattini subgroup in...

A. Brumer has shown that every profinite group of strict cohomological p-dimension 2 possesses a class field theory the tautological class field theory. In particular, this result also applies to the universal p-Frattini extension G̃p of a finite group G. We use this fact in order to establish a class field theory for every p-Frattini extension π : G̃ → G (Thm.A). The role of the class field modu...

A pro-p group G is called strongly Frattini-resistant if the function H↦Φ(H), from poset of all closed subgroups into itself, a embedding. groups appear naturally in Galois theory. Indeed, every maximal over field that contains primitive pth root unity (and also −1 p=2) Frattini-resistant. Let G1 and G2 be non-trivial groups. We prove G1×G2 only one direct factors or torsion-free abelian other ...

‎In 1970‎, ‎Menegazzo [Gruppi nei quali ogni sottogruppo e intersezione di sottogruppi massimali‎, ‎ Atti Accad‎. ‎Naz‎. ‎Lincei Rend‎. ‎Cl‎. ‎Sci‎. ‎Fis‎. ‎Mat‎. ‎Natur. 48 (1970)‎, ‎559--562.] gave a complete description of the structure of soluble $IM$-groups‎, ‎i.e.‎, ‎groups in which every subgroup can be obtained as intersection of maximal subgroups‎. ‎A group $G$ is said to have the $FM$...