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 $r\ge 2$ acting on a finite $q^{\mathrm{\prime }}$-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 $a\in A^{\mathrm{\#}}$, then ${\gamma }_{r-1}(G)$ is $k$-Engel for some $\{n,q,r\}$-bounded number $k$, and if, for some integer $d$ such that $2^d\leq r-1$,allelementsinthe$d$thderivedgroupof$C_G(a)$are$n$-Engelin$G$forany$a\in A^#$,thenthe$d$thderivedgroup$G^{(d)}$is$k$-Engel for some $\{n,q,r\}$-bounded number $k$. Assuming $r\ge 3$ we prove that if all elements in ${\gamma }_{r-2}(C_G(a))$ are $n$-Engel in $C_G(a)$ for any $a\in A^{\mathrm{\#}}$, then ${\gamma }_{r-2}(G)$ is $k$-Engel for some $\{n,q,r\}$-bounded number $k$, and if, for some integer $d$ such that $2^d\le r-2$, all elements in the $d$th derived group of $C_G(a)$ are $n$-Engel in $C_G(a)$ for any $a\in A^{\mathrm{\#}},$ 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.