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SUMMARY:Gareth Tracey (University of Oxford)
DTSTART:20201203T160000Z
DTEND:20201203T170000Z
DTSTAMP:20260423T053048Z
UID:GOThIC/9
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/GOThIC/9/">O
 n the Fitting height and insoluble length of a finite group</a>\nby Gareth
  Tracey (University of Oxford) as part of GOThIC - Ischia Online Group The
 ory Conference\n\n\nAbstract\nA classical result of Baer states that an el
 ement x of a finite group $G$ is contained in the Fitting subgroup $F(G)$ 
 of $G$ if and only if $x$ is a left Engel element of $G$. That is\, $x$ is
  in $F(G)$ if and only if there exists a positive integer $k$ such that $[
 g\, x\, ...\, x]$ (with $x$ appearing $k$ times\, and using the convention
  $[x_1\, x_2\, x_3\, \\dots\, x_k] := [[\\dots [[x_1\, x_2]\, x_3]\, ...]\
 , x_k])$ is trivial for all $g$ in $G$. The result was generalised by E. K
 hukhro and P. Shumyatsky in a 2013 paper via an analysis of the sets $E(G(
 k))= \\{[g\, x\, ...\, x]: g \\in G\\}$.\n\nIn this talk\, we will continu
 e to study the properties of these sets\, concluding with the proof of two
  conjectures made in said paper. As a by-product of our methods\, we also 
 prove a generalisation of a result of Flavell\, which itself generalises W
 ielandt's Zipper Lemma and provides a characterisation of subgroups contai
 ned in a unique maximal subgroup. We also derive a number of consequences 
 of our theorems\, including some applications to the set of odd order elem
 ents of a nite group inverted by an involutory automorphism. Joint work wi
 th R.M. Guralnick.\n
LOCATION:https://researchseminars.org/talk/GOThIC/9/
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