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

TL;DR

This paper extends Lewis's results on a central series related to the vanishing off subgroup of a group, establishing new relations between subgroup indices and central series terms.

Contribution

It generalizes Lewis's findings to higher terms in the series, revealing new equalities involving subgroup indices and central series components.

Findings

Established that for i ≥ 4, |G_{i-1}:V_{i-1}| equals |G:D_{3}|.

Extended the relation between subgroup indices to higher terms in the central series.

Provided conditions under which certain subgroup index equalities hold.

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 $|G:V_{1}|=|G':V_{2}|^{2}=p^{2n}$. Let $D_{3}/V_{3} = C_{G/V_{3}}(G'/V_{3})$. 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 $|G_{i-1}:V_{i-1}|=|G:D_{3}|.$