### Abstract

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