Abstract
Abstract Let q be a prime, n a positive integer and A an elementary abelian group of order q r {q^{r}} with r ≥ 2 {r\geq 2} acting on a finite q ′ {q^{\prime}} -group G. We show that if all elements in γ r - 1 ( C G ( a ) ) {\gamma_{r-1}(C_{G}(a))} are n-Engel in G for any a ∈ A # {a\in A^{\#}} , then γ r - 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 2 d ≤ r - 1 {2^{d}\leq r-1} , all elements in the dth derived group of C G ( a ) {C_{G}(a)} are n-Engel in G for any a ∈ A # {a\in A^{\#}} , then the dth 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 γ r - 2 ( C G ( a ) ) {\gamma_{r-2}(C_{G}(a))} are n-Engel in C G ( a ) {C_{G}(a)} for any a ∈ A # {a\in A^{\#}} , then γ r - 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 2 d ≤ r - 2 {2^{d}\leq r-2} , all elements in the dth derived group of C G ( a ) {C_{G}(a)} are n-Engel in C G ( a ) {C_{G}(a)} for any a ∈ A # , {a\in A^{\#},} then the dth 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.
Highlights
Let A be a finite group acting on a finite group G
Assuming r ≥ 3 we prove that if all elements in γr−2(CG(a)) are n-Engel in CG(a) for any a ∈ A#, γr−2(G) is k-Engel for some {n, q, r}-bounded number k, and if, for some integer d such that 2d ≤ r − 2, all elements in the dth derived group of CG(a) are n-Engel in CG(a) for any a ∈ A#, the dth derived group G(d) is k-Engel for some {n, q, r}-bounded number k
Following the solution of the restricted Burnside problem it was discovered that the exponent of CG(A) may have strong impact over the exponent of G
Summary
Let A be a finite group acting on a finite group G. Let us denote by γi(H) the ith term of the lower central series of a group H and by H(i) the ith term of the derived series of H It was shown in [9] and further in [1, 2] that if the rank of the acting group A is big enough, results of similar nature to that of Theorem 1.1 can be obtained while imposing conditions on elements of γi(CG(a)) or CG(a)(i) rather than on elements of CG(a). Let q be a prime, n a positive integer and A an elementary abelian group of order qr with r ≥ 3 acting coprimely on a profinite group G. If A is a noncyclic abelian group acting coprimely on a profinite group G, G is generated by the subgroups CG(B), where A/B is cyclic
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