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New Facts about the Vanishing Off Subgroup
$V(G)$
- Part of
-
- Journal:
- Canadian Mathematical Bulletin / Volume 63 / Issue 2 / June 2020
- Published online by Cambridge University Press:
- 23 August 2019, pp. 262-268
- Print publication:
- June 2020
-
- Article
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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 
$|G_{i-1}\,:\,V_{i-1}|=|G\,:\,D_{3}|$.
