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

TL;DR

This paper investigates automorphisms of categories of finitely generated free groups within a specific subvariety, providing an example where the quotient of automorphism groups is non-trivial, addressing a question in universal algebraic geometry.

Contribution

It constructs an example of a subvariety of groups where the automorphism quotient group A/Y is non-trivial, answering a posed open question.

Findings

Example of a subvariety with non-trivial A/Y group

Addresses Plotkin's question on automorphism groups

Supports Tsurkov's hypothesis about periodic 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.