Automorphisms of the category of finitely generated free groups of the some subvariety of the variety of all groups

Abstract

In universal algebraic geometry the category of the finite generated free algebras of some fixed variety of algebras and the quotient group A/Y are very important. Here A is a group of all automorphisms of this category and Y is a group of all inner automorphisms of this category. In the varieties of all the groups, all the abelian groups (see B. Plotkin and G. Zhitomirski, 2006), all the nilpotent groups of the class no more then n (see A. Tsurkov, 2007) the group A/Y is trivial. B. Plotkin posed a question: "Is there a subvariety of the variety of all the groups, such that the group A/Y in this subvariety is not trivial?" A. Tsurkov hypothesized that exist some varieties of periodic groups, such that the groups A/Y in these varieties is not trivial. In this paper we give an example of one subvariety of this kind.