Given a positive integer m and a group-word w, we consider a finite group G such that w(G) ≠ 1 and all centralizers of non-trivial w-values have order at most m. We prove that if w=v(x1q1,⋯,xkqk), where v is a multilinear commutator word and q1, ⋯ , qk are p-powers for some prime p, then the order of G is bounded in terms of w and m only. Similar results hold when w is the nth Engel word or the word w= [xn, y1, ⋯ , yk] with k≥ 1.
Finite groups with small centralizers of word-values / Detomi E.; Morigi M.; Shumyatsky P.. - In: MONATSHEFTE FÜR MATHEMATIK. - ISSN 0026-9255. - STAMPA. - 191:2(2020), pp. 257-265. [10.1007/s00605-019-01292-8]
Finite groups with small centralizers of word-values
Morigi M.;
2020
Abstract
Given a positive integer m and a group-word w, we consider a finite group G such that w(G) ≠ 1 and all centralizers of non-trivial w-values have order at most m. We prove that if w=v(x1q1,⋯,xkqk), where v is a multilinear commutator word and q1, ⋯ , qk are p-powers for some prime p, then the order of G is bounded in terms of w and m only. Similar results hold when w is the nth Engel word or the word w= [xn, y1, ⋯ , yk] with k≥ 1.| File | Dimensione | Formato | |
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DMS2020_Small_centralizers_revised.pdf
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