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Profinite groups and centralizers of coprime automorphisms whose elements are Engel
| Content Provider | Semantic Scholar |
|---|---|
| Author | Acciarri, Cristina Silveira, Danilo Sanção Da |
| Copyright Year | 2017 |
| 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. |
| Starting Page | 485 |
| Ending Page | 509 |
| Page Count | 25 |
| File Format | PDF HTM / HTML |
| DOI | 10.1515/jgth-2018-0001 |
| Volume Number | 21 |
| Alternate Webpage(s) | https://arxiv.org/pdf/1707.06889v1.pdf |
| Alternate Webpage(s) | https://doi.org/10.1515/jgth-2018-0001 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
