A Central series associated with the vanishing off subgroup V(G)

Abstract

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 $V_i=[V_{i-1},G]$ for the terms in this central series. Lewis proved that there exists a positive integer $n$ such that if $V_3<G_3$, then $|G:V_1|=|G^{\prime }:V_2{|}^2=p^{2n}$. Let $D_3/V_3=C_{G/V_3}(G^{\prime }/V_3)$. He also showed that if $V_3<G_3$, then either $|G:D_3|=p^n$ or $D_3=V_1$. We show that if $V_i<G_i$ for $i\ge 4,$ where $G_i$ is the $i$-th term in the lower central series of $G$, then $|G_{i-1}:V_{i-1}|=|G:D_3|.$