Fox Derivatives: a Unique Connection Between Group Presentations and the Integral Group Ring

We give a criterion to determine whether generators can be removed from a finite presentation via Tietze transformations. We prove that for a generator in a presentation < X vertical bar R > to be removable, there must exist a word in the normal closure of relators, (R) over bar, whose Fox derivative is an invertible element in ZG. Furthermore, in this case all elements of ZG can be written as the derivative of words in R, with respect to the removable generator. We further discuss the application of this result on the theory units of group rings.