Solvable groups with 𝜋-isolators

Abstract

Let π \pi be any nonempty set of prime numbers. A natural number is a π \pi -number precisely if all of its prime factors are in π \pi . A group G G is said to have the π \pi -isolator property if for every subgroup H H of G G , the set π H = { g ∈ G ; g n ∈ H ^\pi \sqrt H = \{ g \in G;{g^n} \in H is a subgroup of G G . It is well known that nilpotent groups have the π \pi -isolator property for any nonempty set π \pi of primes. Finitely generated solvable linear groups with finite Prüfer rank, and in particular polycyclic groups, have subgroups of finite index with the π \pi -isolator property if π \pi is the set of all primes. It is shown here that if π \pi is any finite nonempty set of primes and G G is a finitely generated solvable group, then G G has a subgroup of finite index with the π \pi -isolator property if and only if G G is nilpotent-by-finite.