On the cardinality of Extremally Disconnected Groups with Linear Topology

A group topology is said to be linear if open subgroups form a base of neighborhoods of the identity element. It is proved that the existence of a nondiscrete extremally disconnected group of Ulam nonmeasurable cardinality with linear topology implies that of a nondiscrete extremally disconnected group of cardinality at most $2^\omega$ with linear topology.