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SUMMARY:Dan Segal (University of Oxford)
DTSTART:20210617T150000Z
DTEND:20210617T160000Z
DTSTAMP:20260423T021321Z
UID:GOThIC/35
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/GOThIC/35/">
 Groups\, Rings\, Logic</a>\nby Dan Segal (University of Oxford) as part of
  GOThIC - Ischia Online Group Theory Conference\n\n\nAbstract\nIn group th
 eory\, interesting statements about a group usually can’t be ex-\npresse
 d in the language of first-order logic. It turns out\, however\, that some
 \ngroups can actually be determined by their first-order properties\, or\,
  even more\nstrongly\, by a single first-order sentence. In the latter cas
 e the group is said to\nbe finitely axiomatizable.\n\nI will describe some
  examples of this phenomenon (joint work with A. Nies\nand K. Tent). One f
 amily of results concerns axiomatizability of $p$-adic analytic\npro-$p$ g
 roups\, within the class of all profinite groups.\n\nAnother main result i
 s that for an adjoint simple Chevalley group of rank at\nleast $2$ and an 
 integral domain $R$\, the group $G(R)$ is bi-interpretable with the\nring 
 $R$. This means in particular that first-order properties of the group $G(
 R)$\ncorrespond to first-order properties of the ring $R$. As many rings a
 re known to\nbe finitely axiomatizable we obtain the corresponding result 
 for many groups\;\nthis holds in particular for every finitely generated g
 roup of the form $G(R)$.\n
LOCATION:https://researchseminars.org/talk/GOThIC/35/
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