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CATEGORIES:Algebra Seminar
SUMMARY:Almost Engel compact groups - Evgeny Khukhro (Cha
rlotte Scott Research Centre for Algebra\, Univers
ity of Lincoln\, UK)
DTSTART:20190328T150000Z
DTEND:20190328T160000Z
UID:TALK3582AT
URL:/talk/index/3582
DESCRIPTION:We say that a group $G$ is almost Engel if for eve
ry $g\\in G$ there is a finite set ${\\mathscr E}
(g)$ such that for every $x\\in G$ all sufficientl
y long commutators $[...[[x\,g]\,g]\,\\dots \,g]$
belong to ${\\mathscr E}(g)$\, that is\, for every
$x\\in G$ there is a positive integer $n(x\,g)$ s
uch that $[...[[x\,g]\,g]\,\\dots \,g]\\in {\\math
scr E}(g)$ if $g$ is repeated at least $n(x\,g)$ t
imes. (Thus\, Engel groups are precisely the almos
t Engel groups for which we can choose ${\\mathscr
E}(g)=\\{ 1\\}$ for all $g\\in G$.)\n\nWe prove t
hat if a compact (Hausdorff) group $G$ is almost E
ngel\, then\n$G$ has a finite normal subgroup $N$
such that $G/N$ is locally nilpotent. If in additi
on there is a uniform bound $|{\\mathscr E}(g)|\\l
eq m$ for the orders of the corresponding sets\, t
hen the subgroup $N$ can be chosen of order bounde
d in terms of $m$. The proofs use the Wilson--Zelm
anov theorem saying that Engel profinite groups ar
e locally nilpotent.\n\nThis is joint work with Pa
vel Shumyatsky.
LOCATION:Nuffield G13
CONTACT:Chris Parker
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