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For an element $g$ of a group $G$, an Engel sink is a subset $mathscr{E}(g)$ such that for every $ xin G $ all sufficiently long commutators $ [x,g,g,ldots,g] $ belong to $mathscr{E}(g)$. Let $q$ be a prime, let $m$ be a positive integer and $A$ an elementary abelian group of order $q^2$ acting coprimely on a finite group $G$. We show that if for each nontrivial element $a$ in $ A$ and every element $gin C_{G}(a)$ the cardinality of the smallest Engel sink $mathscr{E}(g)$ is at most $m$, then the order of $gamma_infty(G)$ is bounded in terms of $m$ only. Moreover we prove that if for each $ain Asetminus {1}$ and every element $gin C_{G}(a)$, the smallest Engel sink $mathscr{E}(g)$ generates a subgroup of rank at most $m$, then the rank of $gamma_infty(G)$ is bounded in terms of $m$ and $q$ only.
For an element $g$ of a group $G$, an Engel sink is a subset $mathcal{E}(g)$ such that for every $ xin G $ all sufficiently long commutators $ [x,g,g,ldots,g] $ belong to $mathcal{E}(g)$. We conjecture that if $G$ is a profinite group in which every
The main result of the paper is the following theorem. Let $q$ be a prime and $A$ an elementary abelian group of order $q^3$. Suppose that $A$ acts coprimely on a profinite group $G$ and assume that $C_G(a)$ is locally nilpotent for each $ain A^{#}$. Then the group $G$ is locally nilpotent.
Let $G$ be a group and let $xin G$ be a left $3$-Engel element of order dividing $60$. Suppose furthermore that $langle xrangle^{G}$ has no elements of order $8$, $9$ and $25$. We show that $x$ is then contained in the locally nilpotent radical of $G
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 $G$. The following results are proved. We show that if all elements in $gamma_{r-1}(C_G(a))$ are $n$-Engel in $G
If K is a discrete group and Z is a K-spectrum, then the homotopy fixed point spectrum Z^{hK} is Map_*(EK_+, Z)^K, the fixed points of a familiar expression. Similarly, if G is a profinite group and X is a discrete G-spectrum, then X^{hG} is often gi