On the number of normal subgroups of an uncountable group

Abstract

AbstractIn this paper two theorems are proved that give a partial answer to a question posed by G. Behrendt and P. Neumann. Firstly, the existence of a group of cardinality ℵ1 with exactly ℵ1 normal subgroups, yet having a subgroup of index 2 with 2ℵ1 normal subgroups, is consistent with ZFC (the Zermelo-Fraenkel axioms for set theory together with the Axiom of Choice). Secondly, the statement “Every metabelian-by-finite group of cardinality ℵ1 has 2ℵ1 normal subgroups” is consistent with ZFC.