The algorithms that recognize Milnor laws and properties of these laws

Abstract

We consider several equivalent definitions of the so-called Milnor
laws (or Milnor identities) that is the laws which are not satisfied in \(\mathfrak{A}_p\mathfrak{A}\) varieties. The purpose of this article is to provide algorithms that allow us to check whether a given identity \(w(x,y)\) has one of the following properties:

\(w(x,y)\) is a Milnor law,

every nilpotent group satisfying \(w(x,y)\) is abelian,

every finitely generated metabelian group satisfying \(w(x,y)\) is finite-by-abelian.