Let $q$ be a prime, $n$ a positive integer and $A$ an elementary abelian group of order $q^r$ with $rgeq2$ acting on a finite $q'$-group $G$. We show that if all elements in $gamma_{r-1}(C_G(a))$ are $n$-Engel in $G$ for any $ain A^#$, then $gamma_{r-1}(G)$ is $k$-Engel for some ${n,q,r}$-bounded number $k$, and if, for some integer $d$ such that $2^dleq r-1$, all elements in the $d$th derived group of $C_G(a)$ are $n$-Engel in $G$ for any $ain A^#$, then the $d$th derived group $G^{(d)}$ is $k$-Engel for some ${n,q,r}$-bounded number $k$. Assuming $rgeq 3$ we prove that if all elements in $gamma_{r-2}(C_G(a))$ are $n$-Engel in $C_G(a)$ for any $ain A^#$, then $gamma_{r-2}(G)$ is $k$-Engel for some ${n,q,r}$-bounded number $k$, and if, for some integer $d$ such that $2^dleq r-2$, all elements in the $d$th derived group of $C_G(a)$ are $n$-Engel in $C_G(a)$ for any $ain A^#,$ then the $d$th derived group $G^{(d)}$ is $k$-Engel for some ${n,q,r}$-bounded number $k$. Analogous (non-quantitative) results for profinite groups are also obtained.

Profinite groups and centralizers of coprime automorphisms whose elements are Engel / Acciarri, C; da Silveira, D S. - In: JOURNAL OF GROUP THEORY. - ISSN 1433-5883. - 21:(2018), pp. 485-509. [10.1515/jgth-2018-0001]

Profinite groups and centralizers of coprime automorphisms whose elements are Engel

Acciarri C;
2018-01-01

Abstract

Let $q$ be a prime, $n$ a positive integer and $A$ an elementary abelian group of order $q^r$ with $rgeq2$ acting on a finite $q'$-group $G$. We show that if all elements in $gamma_{r-1}(C_G(a))$ are $n$-Engel in $G$ for any $ain A^#$, then $gamma_{r-1}(G)$ is $k$-Engel for some ${n,q,r}$-bounded number $k$, and if, for some integer $d$ such that $2^dleq r-1$, all elements in the $d$th derived group of $C_G(a)$ are $n$-Engel in $G$ for any $ain A^#$, then the $d$th derived group $G^{(d)}$ is $k$-Engel for some ${n,q,r}$-bounded number $k$. Assuming $rgeq 3$ we prove that if all elements in $gamma_{r-2}(C_G(a))$ are $n$-Engel in $C_G(a)$ for any $ain A^#$, then $gamma_{r-2}(G)$ is $k$-Engel for some ${n,q,r}$-bounded number $k$, and if, for some integer $d$ such that $2^dleq r-2$, all elements in the $d$th derived group of $C_G(a)$ are $n$-Engel in $C_G(a)$ for any $ain A^#,$ then the $d$th derived group $G^{(d)}$ is $k$-Engel for some ${n,q,r}$-bounded number $k$. Analogous (non-quantitative) results for profinite groups are also obtained.
21
485
509
Profinite groups and centralizers of coprime automorphisms whose elements are Engel / Acciarri, C; da Silveira, D S. - In: JOURNAL OF GROUP THEORY. - ISSN 1433-5883. - 21:(2018), pp. 485-509. [10.1515/jgth-2018-0001]
Acciarri, C; da Silveira, D S
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11380/1255528
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