6533b7d0fe1ef96bd125b9ed

RESEARCH PRODUCT

Saturated formations and products of connected subgroups

subject

description

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-called nilpotent-like Fitting formations of soluble groups, and to finite groups that are products of arbitrarily many pairwise permutable F -connected F -subgroups.