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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:On blocks with few irreducible characters - Noelia
Rizo (Universidad de Oviedo)
DTSTART;TZID=Europe/London:20220509T115000
DTEND;TZID=Europe/London:20220509T122000
UID:TALK172775AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/172775
DESCRIPTION:W. Burnside proposed to characterize finite groups
with a given number of irreducible characters. Th
e block-wise version of this problem is to charact
erize the defect groups of a $p$-block $B$ of a fi
nite group $G$ with a given number of irreducible
characters in it.In this context\, R. Brauer prove
d that $k(B)=1$ if\, and only if\, the defect grou
p of $B$ is trivial. Years later\, J. Brandt showe
d that $k(B)=2$ if\, and only if\, the defect grou
ps of $B$ are cyclic of order 2. These two results
do not require the Classification of Finite Simpl
e Groups. However\, the complexity of this problem
seems to explode when we deal with the next situa
tion\, namely when we try to classify the defect g
roups of $p$-blocks satisfying $k(B)=3$. It is con
jectured that in this case the defect groups of $B
$ are cyclic of order 3. When $B$ is the principal
block or $D$ is normal in $G$\, the situation is
much better understood and the conjecture is known
to hold in this case.If $k(B)=4$ and $B$ is the p
rincipal block\, then S. Koshitani and T. Sakurai
have proven that $|D|=4$ or 5. We show that the sa
me is obtained whenever $B$ is an arbitrary block
of $G$ and $D$ is normal in $G$. Moreover\, we dea
l with the next natural situation\, namely where $
k(B)=5$ and $B$ is the principal block of $G$\, in
which case we obtain that $D\\cong {\\sf C}_5\,{\
\sf C}_7\, {\\sf D}_8\, {\\sf Q}_8$.This talk is a
n overview of joint works with J.M. Martí\;n
ez\, L. Sanus\, M. Schaeffer Fry and C. Vallejo.
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:
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