Torsion elements in \(p\)-adic analytic pro-\(p\) groups. (Q1882894)

A finitely generated pro-\(p\) group \(G\) is \(p\)-adic analytic if and only if there exist \(m\) and \(h\), with \(m<p^h\), such that the \(m\)-th term \(\gamma_m(G)\) of the lower central series of \(G\) is contained in \(G^{p^h}\). A result from [Analytic pro-\(p\) groups (Cambridge University Press) (1999; Zbl 0934.20001)] shows that if \(m=2\), then the torsion elements in \(G\) form a subgroup. A result of \textit{J.~González} and the author [submitted for publication] shows that this holds for \(m\leq 2p-3\). (This is an empty statement for \(p=2\).) In the same paper examples are given of groups \(G\) in which \(m=2p-1\), and the torsion elements do not form a subgroup. Also, in these examples for all \(h\geq 2\) one has \(\gamma_{h(p-1)+1}(G)\leq G^{p^h}\). In the paper under review the remaining case \(m=2p-2\) is dealt with, showing in the meantime that these examples are best possible. The main result is the following. Let \(p\) be an odd prime, and \(G\) a finitely generated pro-\(p\) group. If \(h\geq 1\), and \(\gamma_{h(p-1)}(G)\leq G^{p^h}\), then the torsion elements of \(G\) form a subgroup. An explicit bound on the order of the product of two elements is provided.