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 $r\geq2$ acting on a finite $q'$-group $G$. The following results are proved. We show that if all elements in $\gamma_{r-1}(C_G(a))$ are $n$-Engel in $G$ for any $a\in A^\#$, then $\gamma_{r-1}(G)$ is $k$-Engel for some $\{n,q,r\}$-bounded number $k$, and if, for some integer $d$ such that $2^d\leq r-1$, all elements in the $d$th derived group of $C_G(a)$ are $n$-Engel in $G$ for any $a\in A^\#$, then the $d$th derived group $G^{(d)}$ is $k$-Engel for some $\{n,q,r\}$-bounded number $k$. Assuming $r\geq 3$ we prove that if all elements in $\gamma_{r-2}(C_G(a))$ are $n$-Engel in $C_G(a)$ for any $a\in A^\#$, then $\gamma_{r-2}(G)$ is $k$-Engel for some $\{n,q,r\}$-bounded number $k$, and if, for some integer $d$ such that $2^d\leq r-2$, all elements in the $d$th derived group of $C_G(a)$ are $n$-Engel in $C_G(a)$ for any $a\in A^\#,$ then the $d$th derived group $G^{(d)}$ is $k$-Engel for some $\{n,q,r\}$-bounded number $k$. Analogue (non-quantitative) results for profinite groups are also obtained.