Profinite groups with few conjugacy classes of elements of infinite order (Q6156368)

A classical result by \textit{E. I. Zelmanov} [Isr. J. Math. 77, No. 1--2, 83--95 (1992; Zbl 0786.22008)] states that a periodic profinite group is locally finite. It is well-known that an infinite profinite group is uncountable. Moreover in [Proc. Am. Math. Soc. 147, No. 9, 4083--4089 (2019; Zbl 1425.22004)], \textit{A. Jaikin-Zapirain} and \textit{N. Nikolov} proved that if a profinite group has only countably many conjugacy classes, then it is finite. On the other hand, the author also proved in [\textit{J. S. Wilson}, Proc. Am. Math. Soc. 150, No. 8, 3297--3305 (2022; Zbl 1498.20072)] that a profinite group has countably many conjugacy classes of \(p\)-elements for an odd prime \(p\) if and only if its Sylow \(p\)-subgroups are finite. In this paper, inspired by these results, the author proves the following deep theorem: Theorem A. If \(G\) is a (topologically) finitely generated profinite group with countably many conjugacy glasses of elements of infinite order, then \(G\) is finite.