Equations solvable by radicals in a uniquely divisible group

TL;DR

This paper investigates the solvability of certain equations in uniquely divisible groups, introducing polynomial criteria to classify which equations are solvable by radicals and identifying new non-solvable cases.

Contribution

It provides the first infinite families of word equations in such groups that are proven not solvable by radicals, along with a polynomial-based classification conjecture.

Findings

Identified infinite families of non-solvable equations by radicals.

Developed polynomial criteria involving irreducibility for solvability.

Proposed a conjecture for complete classification of solvable equations.

Abstract

We study equations in groups G with unique m-th roots for each positive integer m. A word equation in two letters is an expression of the form w(X,A) = B, where w is a finite word in the alphabet {X,A}. We think of A,B in G as fixed coefficients, and X in G as the unknown. Certain word equations, such as XAXAX=B, have solutions in terms of radicals, while others such as XXAX = B do not. We obtain the first known infinite families of word equations not solvable by radicals, and conjecture a complete classification. To a word w we associate a polynomial P_w in Z[x,y] in two commuting variables, which factors whenever w is a composition of smaller words. We prove that if P_w(x^2,y^2) has an absolutely irreducible factor in Z[x,y], then the equation w(X,A)=B is not solvable in terms of radicals.