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 $FM$...

Abstract We construct a 2-generated pro-2 group with full normal Hausdorff spectrum [ 0 , 1 stretchy="false">] [0,1] , respect to each of the four standard filtration series: 2-power series, lower 2-series, Frattini and dimension subgroup...

‎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 $fm$...

Let G be a finite group of even order v. Does there exist a 1−factorization of Kv admitting G as an automorphism group acting sharply transitively on vertices? If G is cyclic and v = 2, for t ≥ 3, then the answer to the previous question is known to be negative by a result of A.Hartman and A.Rosa (1985). For several large families of groups of even order constructions have always been found thu...

Consider a finite group G and subgroups H,K of G. We say that H and K permute if HK = KH and call H a permutable subgroup if H permutes with every subgroup of G. A group G is called quasi-Dedekind if all subgroups of G are permutable. We can define, for every finite group G, an arithmetic quantity that measures the probability that two subgroups (chosen uniformly at random with replacement) per...