{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:CM4IRZ7CMELTJP2L6YGRZHBYW5","short_pith_number":"pith:CM4IRZ7C","schema_version":"1.0","canonical_sha256":"133888e7e2611734bf4bf60d1c9c38b74bcfa9b104a055dd8f018cb0d018ef3b","source":{"kind":"arxiv","id":"1209.2886","version":1},"attestation_state":"computed","paper":{"title":"A Central series associated with the vanishing off subgroup V(G)","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Nabil Mlaiki","submitted_at":"2012-09-13T13:33:10Z","abstract_excerpt":"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 $|G:V_{1}|=|G':V_{2}|^{2}=p^{2n}$. Let $D_{3}/V_{3} = C_{G/V_{3}}(G'/V_{3})$. 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}