This paper contains a list of crucial mistakes and counterexamples to some of the main statements in the paper «Elementary theory of free nonabelian groups» by O. Kharlampovich and A. Myasnikov, which was published in the Journal of Algebra in June 2006.

This article is dedicated to the investigation of difficulties involved in the understanding of the homomorphism concept. It doesn’t restrict to group-theory but on the contrary raises the issue of developing teaching strategies aiming at gaining access to structuralist thinking. Emphasis is put on epistemological analysis and its interaction with didactics in an attempt to make Abstract Algebra more accessible.

This is a survey of methods developed in the last few years to prove results on growth in non-commutative groups. These techniques have their roots in both additive combinatorics and group theory, as well as other fields. We discuss linear algebraic groups, with SL_2(Z/pZ) as the basic example, as well as permutation groups. The emphasis lies on the ideas behind the methods.