Canadian Mathematical Bulletin ( IF 0.5 ) Pub Date : 2019-08-23 , DOI: 10.4153/s0008439519000195 Nabil Mlaiki
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 $|G_{i-1}\,:\,V_{i-1}|=|G\,:\,D_{3}|$ .
中文翻译:
有关消失的子组 $ V(G)$的 新事实
在本手稿中,我们概括了刘易斯关于与消失的子组相关的中心序列的结果。我们为消失的 $ G $ 子组 写 $ V_ {1} = V(G)$ ,并为该中心系列中的术语写 $ V_ {i} = [V_ {i-1},G] $ 。Lewis证明存在一个正整数 $ n $ ,使得如果 $ V_ {3} <G_ {3} $ ,则 $ | G \,:: \,V_ {1} | = | G ^ {\ prime} \, :\,V_ {2} | ^ {2} = p ^ {2n} $ 。设 $ D_ {3} / V_ {3} = C_ {G / V_ {3}}(G ^ {\ prime} / V_ {3})$ 。他还表明,如果 $ V_ {3} <G_ {3} $ ,则 $ | G \,:\,D_ {3} | = p ^ {n} $ 或 $ D_ {3} = V_ {1} $ 。我们证明如果 $ V_ {i} <G_ {i} $ 对于 $ i \ geqslant 4 $ ,其中 $ G_ {i} $ 是 $ G $ 的较低中央序列中的 $ i $ -th项 ,然后 $ | G_ {i-1} \,:\,V_ {i -1} | = | G \,:\,D_ {3} | $ 。