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Seminario / Tracey

The organising committee of the Ischia Online Group Theory Conference
(GOThIC) is inviting you to a scheduled Zoom meeting.


- The TIME OF THE TALK is 17:00 CET = UTC + 1.
- You  are  welcome to  share  the  Zoom  link with  other  interested

- When joining, please  MAKE SURE THAT YOUR NICKNAME IS  
YOUR NAME AND SURNAME, or  close to it, so  that the organisers
can  recognise you and let you in

The Ischia Group Theory 2020 Conference
(http://www.dipmat2.unisa.it/ischiagrouptheory/) was planned
for 30 March - 4 April 2020. It has now been postponed. In the
meantime, we are offering a series of online lectures by leading
researchers (https://sites.google.com/unisa.it/e-igt2020/).

TIME: December 3rd, 2020 17:00 CET (UTC+1)

COFFEE BREAK: The  talk will start at 17:00 CET. 
The conference room  will open at 16:45 CET for a coffee break
- Bring Your Own tea/coffee mug - biscuits appreciated - and join us
for some small talk before the event.
SPEAKER: Gareth Tracey (University of Oxford)
TITLE: On the Fitting height and insoluble length of a finite group


A classical result of Baer states that an element 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 [x1, x2, x3, ..., xk] := [[... [[x1, x2], x3], ...], xk]) is trivial for all g in G. The result was generalised by E. Khukhro and P. Shumyatsky in a 2013 paper via an analysis of the sets E(G(k))= {[g, x, ..., x]: g in G}.

In this talk, we will continue 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 Wielandt's Zipper Lemma and provides a characterisation of subgroups contained in a unique maximal subgroup. We also derive a number of consequences of our theorems, including some applications to the set of odd order elements of a nite group inverted by an involutory automorphism. Joint work with R.M. Guralnick.

02 dicembre 2020
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