Generalised Hopficity and Products of the Integers

Abstract

Hopfian groups have been a topic of interest in algebraic settings for many years. In this work a natural generalization of the notion, so-called R-Hopficity is introduced. Basic properties of R-Hopfian groups are developed and the question of whether or not infinite direct products of copies of the integers are R-Hopfian is considered. An unexpected result is that the answer to this purely algebraic question depends on the set theory assumed.2010 Mathematics Subject Classification. 20K30, 20k20.