The number of homomorphisms from the Hawaiian earring group

Abstract

We show a dichotomy for groups of cardinality less than continuum. The number of homomorphisms from the Hawaiian earring group to such a group $G$ is either the cardinality of $G$ in case $G$ is noncommutatively slender, or the number is $2^{2^{\aleph_0}}$ in case $G$ is not noncommutatively slender. An example of a noncommutatively slender group with nontrivial divisible element is exhibited.