Equations solvable by radicals in a uniquely divisible group
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 Pw in Z[x,y] in two commuting variables, which factors whenever w is a composition of smaller words. We prove that if Pw(x2,y2) has an absolutely irreducible factor in Z[x,y], then the equation w(X,A)=B is not solvable in terms of radicals.