Algorithmic undecidability of recognizing a non-Markovian group property

Abstract

Abstract We prove algorithmic undecidability of the following group property: “the positive theory of a group G coincides with the positive theory of a free non-Abelian group” or (which is equivalent) “the positive theory of a group G coincides with the positive theory of the class of all groups”. This group property is non-Markovian, therefore its algorithmic undecidability does not follow from the fundamental Adyan–Rabin theorem.