Coloquios del departamento 2019
Próxima charla: Bob Oliver "Local structure of finite groups and of their classifying spaces."  Jueves 14 de marzo de 2019  15h  Aula de seminario nuevo del DM/IMAS.

Coloquios del Departamento de Matemática 2019 
Coloquios 2019
jueves 14 de marzo de 2019  15h  Aula de seminario nuevo del DM/IMAS.
Bob Oliver (Université Paris 13  Francia)
Local structure of finite groups and of their classifying spaces.
Resumen:
Let p be a prime. By the plocal structure of a finite group G
is meant a Sylow psubgroup S and the Gconjugacy relations among the
subgroups of S. More precisely, two finite groups G and H are said to be
plocally equivalent (have the same plocal structure) if there is an
isomorphism S>T for some S in Syl_p(G) and T in Syl_p(H), that
preserves all G and Hconjugacy relations among subgroups of S and of T.
A classifying space of a finite group G is an EilenbergMacLane space of
type K(G; 1); i.e., a topological space BG whose fundamental group is
isomorphic to G and whose universal covering space is contractible. Two
classifying spaces BG and BH are said to be plocally equivalent if
there is a third space X, and maps f:BG>X and f':BH>X, such that f and
f' both induce isomorphisms in homology with coefficients in Z/p.
A conjecture by Martino and Priddy, now a theorem, says that for each
pair of finite groups G and H and each prime p, G and H are plocally
equivalent if and only if their classifying spaces are plocally
equivalent. This was first proven by me, but only by assuming the
classification of finite simple groups and doing a long, messy
casebycase analysis involving those groups. Later, a much simpler
proof was
found by Chermak, but still assuming the classification of finite simple
groups, and recently Glauberman and Lynd succeeded in modifying
Chermak’s argument to get a classificationfree proof.
In the talk, I want to first describe in more detail the background of
this theorem, and also talk very briefly about some of the ideas behind
its proof. I’ll then give an application: a purely group theoretic
statement whose only known proof is based on homotopy theory. If there’s
time left, I’ll then describe a related result involving automorphisms
of groups and of their classifying spaces, and give a few more examples
and applications involving finite simple groups.



Created by
nsaintie
Last modified
20190314 10:53 AM