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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 $G$. In particular all the left $3$-Engel elements of a group of exponent $60$ are contained in the locally nilpotent radical.
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
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 e
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
A regular left-order on finitely generated group $G$ is a total, left-multiplication invariant order on $G$ whose corresponding positive cone is the image of a regular language over the generating set of the group under the evaluation map. We show th
We study left orderable groups by using dynamical methods. We apply these techniques to study the space of orderings of these groups. We show for instance that for the case of (non-Abelian) free groups, this space is homeomorphic to the Cantor set. W