SOLUTIONS TO ZERO-SUM EXPONENT EQUATIONS OVER FINITE CYCLIC GROUPS OF EXPONENT GREATER THAN TWO

Abstract

Lyndon initiated a study of equations over cyclic groups [8] and his approach led him to consider a class of equations of the form ah= (tpat-p)(tqat-q)(tpat-p)-1(tqat-q)-1where t is the variable and the cyclic group is Cn= 〈a|an〉. Work by Ferlini, Goldstein, and Salpukis [3] concentrated, more generally, on equations with the exponent sum of the variable being zero over C2. This paper continues along the path started by Lyndon and shows that a large class of the equations above do have solutions over Cnwhere n ≥ 4. Our approach involves the use of pictures. We also include a result that the equations with h = 1, 2, p = 1, and q = 2 do not have a solution over C3.