Profinite Groups with Few Conjugacy Classes

Profinite groups arise in many guises in mathematics, notably as the quotients of compact Hausdorff groups by the connected component of the identity. It is well known that the cardinality of an infinite profinite group cannot be less than 2ℵ0, the cardinality of the continuum. In 2019, Jaikin-Zapirain and Nikolov proved moreover that each infinite profinite group has at least 2ℵ0 conjugacy classes. After a brief introduction to profinite groups we discuss the consequences of restricting the number of conjugacy classes of elements of various types (such as p-elements or elements of infinite order). In particular, every finitely generated profinite group with fewer than 2ℵ0 conjugacy classes of elements of infinite order is finite.

Despite the apparent simplicity of the questions, the answers seem to depend on results such as the classification of the finite simple groups (and their automorphism groups) and work of Zel’manov on Lie algebras associated with profinite p-groups.