Profinite groups and centralizers of coprime automorphisms whose elements are Engel

Let q be a prime, n a positive integer and A an elementary abelian group of order q^r with r ≥ 2 acting on a finite q′-group G. We show that if all elements in γ_(r−1)(C_G(a)) are n-Engel in G for any a ∈ A♯, then γ_(r−1)(G) is k-Engel for some {n,q,r}-bounded number k, and if, for some integer d such that 2^d ≤ r−1, all elements in the dth derived group of C_G(a) are n-Engel in G for any a ∈ A♯, then the dth derived group G^(d) is k-Engel for some {n,q,r}-bounded number k. Assuming r ≥ 3 we prove that if all elements in γ_(r−2)(C_G(a)) are n-Engel in C_G(a) for any a ∈ A♯, then γ_(r−2)(G) is k-Engel for some {n,q,r}-bounded number k, and if, for some integer d such that 2^d ≤ r−2, all elements in the dth derived group of C_G(a) are n-Engel in C_G(a) for any a ∈ A♯, then the dth derived group G^(d) is k-Engel for some {n,q,r}-bounded number k. Analogous (non-quantitative) results for profinite groups are also obtained.