Profinite groups with few conjugacy classes of elements of infinite order

TL;DR

This paper proves that finitely generated profinite groups with fewer than continuum many conjugacy classes of infinite order elements must be finite.

Contribution

It establishes a new finiteness criterion for finitely generated profinite groups based on the number of conjugacy classes of infinite order elements.

Findings

Finitely generated profinite groups with less than continuum conjugacy classes of infinite order are finite.

The result links the size of conjugacy class sets to the group's finiteness.

Provides a classification criterion for profinite groups based on conjugacy class cardinality.

Abstract

It is proved that every finitely generated profinite group with fewer than $2^{\aleph_0}$ conjugacy classes of elements of infinite order is finite