Finite groups with small centralizers of word-values

Abstract

Given a positive integer m and a group-word w, we consider a finite group G such that $$w(G) \ne 1$$ and all centralizers of non-trivial w-values have order at most m. We prove that if $$w=v(x_1^{q_1},\dots ,x_k^{q_k})$$, where v is a multilinear commutator word and $$q_1, \dots , q_k$$ 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 $$w=[x^n, y_1, \dots ,y_k]$$ with $$k \ge 1$$.