Localizations of free soluble groups (Q2705013)

Let \(\pi\) be a set of primes and let \(\pi'\) denote the primes not contained in \(\pi\). A group \(G\) is said to be \(\pi\)-local if the map \(x\mapsto x^q\) is bijective for every prime \(q\not\in\pi\). A \(\pi\)-localization of a group \(G\) is a homomorphism \(\phi_\pi\colon G\to G_\pi\), where \(G_\pi\) is a \(\pi\)-local group, with the universal property that given any homomorphism \(\theta\colon G\to H\), with \(H\) \(\pi\)-local, there exists a unique homomorphism \(\theta_\pi\colon G_\pi\to H\) such that \(\theta_\pi\phi_\pi=\theta\).NEWLINENEWLINENEWLINEThe author proves that if \(G\) is a free (non-Abelian) soluble group, then \(G_\pi\) is not soluble. Also, the author gives counterexamples to some problems related to localization.