On the closure of cyclic subgroups of a free group in some profinite topologies

Abstract We determine the closure of a cyclic subgroup 𝐻 of a free group for the pro-𝐕 topology when 𝐕 is an extension-closed pseudovariety of finite groups. We show that 𝐻 is always closed for the pro-nilpotent topology and compute its closure for the pro- <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="bold">G</m:mi> <m:mi>p</m:mi> </m:msub> </m:math> \mathbf{G}_{p} and pro- <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="bold">V</m:mi> <m:mi>p</m:mi> </m:msub> </m:math> \mathbf{V}_{p} topologies, where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="bold">G</m:mi> <m:mi>p</m:mi> </m:msub> </m:math> \mathbf{G}_{p} is the pseudovariety of finite 𝑝-groups and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi mathvariant="bold">V</m:mi> <m:mi>p</m:mi> </m:msub> <m:mo>⊃</m:mo> <m:msub> <m:mi mathvariant="bold">G</m:mi> <m:mi>p</m:mi> </m:msub> </m:mrow> </m:math> \mathbf{V}_{p}\supset\mathbf{G}_{p} denote the pseudovarieties involved in the join decomposition of the pseudovariety of finite supersolvable groups. More generally, given any nonempty set 𝑃 of primes, we consider the stated problem for the pseudovariety <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="bold">G</m:mi> <m:mi>P</m:mi> </m:msub> </m:math> \mathbf{G}_{P} of all finite groups having order a product of primes in 𝑃.