Geodesic
New Facts about the Vanishing Off Subgroup $V(G)$
Canadian mathematical bulletin, Tome 63 (2020) no. 2, pp. 262-268

Voir la notice de l'article provenant de la source Cambridge University Press

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}, 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}, then either $|G\,:\,D_{3}|=p^{n}$ or $D_{3}=V_{1}$. We show that if $V_{i} for $i\geqslant 4$, where $G_{i}$ is the $i$-th term in the lower central series of $G$, then $|G_{i-1}\,:\,V_{i-1}|=|G\,:\,D_{3}|$.
DOI : 10.4153/S0008439519000195
Mots-clés : vanishing off subgroup V (G), lower and upper central series 
Mlaiki, Nabil. New Facts about the Vanishing Off Subgroup $V(G)$. Canadian mathematical bulletin, Tome 63 (2020) no. 2, pp. 262-268. doi: 10.4153/S0008439519000195
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