Seminario / Ersoy
(GOThIC) is inviting you to a scheduled Zoom meeting.
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The Ischia Group Theory 2020 Conference
(http://www.dipmat2.unisa.it/ischiagrouptheory/), planned for
30 March - 4 April 2020, was postponed. In the
meantime, we are organising a series of online lectures by leading
TIME: Thursday February 11th, 2021 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
join us for some smalltalk before the event.
Title: On the centralizer depth in simple locally finite groups
Abstract: Let G be an infinite group. Let Γ_r be the set of all r-generated subgroups of G. The centralizer depth of G (or Cd(G)) is the smallest r such that CG(F) is finitely generated where 1 6= F ∈ Γr. If for every finitely generated subgroup Q of G the centralizer CG(Q) is infinitely generated, then Cd(G) is infinite. For immediate examples: If G is finitely generated and Z(G) 6= 1 then Cd(G) = 1. Let G be one of the 2-generated simple p-groups constructed by A. Olshanskii. Every non-trivial element has a cyclic centralizer, so Cd(G) = 1. We can reformulate the following well-known result in this terminology. Theorem 1. (Hartley-Kuzucuo˘glu) Let G be a simple locally finite group. Then Cd(G) is strictly greater than 1. Remember the following two open questions on centralizers in simple locally finite groups by B. Hartley. A solution for the first one may lead a solution for the second one. Question 2. (Hartley) (1) Does there exist a non-linear simple locally with a linear centralizer of an element?
(2) Does there exist a non-linear simple locally finite group with a finite subgroup F with finite centralizer? In particular, the second question of Hartley asks in every nonlinear simple locally finite group G, is the Cd(G) infinite? We think about this concept since it may give a measure for simple locally finite groups subject to being linear. In particular, the following observation is interesting: Proposition 3. Let G be a linear simple locally finite group. Then Cd(G) = 2. Our aim is to prove the converse, namely we ask the following: Question 4. Let G be a simple locally finite group with Cd(G) = 2. Is G linear? In this talk we will talk about this problem and prove some related results on centralizers in simple locally finite groups. This is a joint project with M. Brescia