Saturated formations and products of connected subgroups

Abstract For a non-empty class of groups C , two subgroups A and B of a group G are said to be C -connected if 〈 a , b 〉 ∈ C for all a ∈ A and b ∈ B . Given two sets π and ρ of primes, S π S ρ denotes the class of all finite soluble groups that are extensions of a normal π-subgroup by a ρ-group. It is shown that in a finite group G = A B , with A and B soluble subgroups, then A and B are S π S ρ -connected if and only if O ρ ( B ) centralizes A O π ( G ) / O π ( G ) , O ρ ( A ) centralizes B O π ( G ) / O π ( G ) and G ∈ S π ∪ ρ . Moreover, if in this situation A and B are in S π S ρ , then G is in S π S ρ . This result is then extended to a large family of saturated formations F , the so-c…

Frattiniduale und Fittingklassen endlicher auflösbarer Gruppen

Injectors and Radicals in Products of Totally Permutable Groups

Abstract Two subgroups H and K of a group G are said to be totally permutable if every subgroup of H permutes with every subgroup of K. In this paper the behaviour of radicals and injectors associated to Fitting classes in a product of pairwise totally permutable finite groups is studied.

�ber Frattiniduale in endlichen Gruppen

Products of pairwise totally permutable groups

[EN] In this paper finite groups factorized as products of pairwise totally permutable subgroups are studied in the framework of Fitting classes

2-Engel relations between subgroups

Abstract In this paper we study groups G generated by two subgroups A and B such that 〈 a , b 〉 is nilpotent of class at most 2 for all a ∈ A and b ∈ B . A detailed description of the structure of such groups is obtained, generalizing the classical result of Hopkins and Levi on 2-Engel groups.

Fitting classes and products of totally permutable groups

The second and third authors have been supported by Proyecto PB 97-0674-C02-02 of DGESIC, Ministerio de Educación y Cultura, Spain.