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New Facts about the Vanishing Off Subgroup 
$V(G)$
Part of:
Representation theory of groups
Published online by Cambridge University Press: 23 August 2019
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 
$|G_{i-1}\,:\,V_{i-1}|=|G\,:\,D_{3}|$.
MSC classification
Primary:
20D10: Solvable groups, theory of formations, Schunck classes, Fitting classes, $pi$-length, ranks
Secondary:
20D15: Nilpotent groups, $p$-groups
- Type
- Article
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- Copyright
- © Canadian Mathematical Society 2019
References
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