Abstract
In this manuscript, we generalize Lewis's result about a central series associated with the vanishing off subgroup. We write V-1 = V(G) for the vanishing off subgroup of G, and V-i = [Vi-1, G] for the terms in this central series. Lewis proved that there exists a positive integer n such that if V-3 < G(3), then vertical bar G : V-1 vertical bar = vertical bar G' : V-2 vertical bar(2) = p(2n). Let D-3/V-3 = C-G/V3 (G'/V-3). He also showed that if V-3 < G(3), then either vertical bar G: D-3 vertical bar = p(n) or D-3 = V-1. We show that if V-i < G(i) for i >= 4, where G(i) is the i-th term in the lower central series of G, then vertical bar G(i-1) : Vi-1 vertical bar = vertical bar G : D-3 vertical bar.