Many non-embeddable infinite groups

Abstract In an elementary way, we construct a family of pairwise non-embeddable torsion-free groups which contains <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mn>2</m:mn> <m:mi>κ</m:mi> </m:msup> </m:math> 2^{\kappa} groups of cardinality 𝜅 for every infinite cardinal number 𝜅 up to the first strong limit cardinal of uncountable cofinality.