  In this paper we study the behavior of the fractions of a factorial design
under permutations of the factor levels. We focus on the notion of regular
fraction and we introduce methods to check whether a given symmetric orthogonal
array can or can not be transformed into a regular fraction by means of
suitable permutations of the factor levels. The proposed techniques take
advantage of the complex coding of the factor levels and of some tools from
polynomial algebra. Several examples are described, mainly involving factors
with five levels.
