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New Facts about the Vanishing Off Subgroup $V(G)$

Published online by Cambridge University Press:  23 August 2019

Nabil Mlaiki*
Affiliation:
Department of Mathematics and General Sciences, Prince Sultan University, Riyadh, Saudi Arabia Email: [email protected]@gmail.com

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 $V_{i}=[V_{i-1},G]$ 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 $i\geqslant 4$, where $G_{i}$ is the $i$-th term in the lower central series of $G$, then $|G_{i-1}\,:\,V_{i-1}|=|G\,:\,D_{3}|$.

Type
Article
Copyright
© Canadian Mathematical Society 2019

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