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SUMMARY:On blocks with few irreducible characters - Noelia Rizo (Universid
 ad de Oviedo)
DTSTART:20220509T105000Z
DTEND:20220509T112000Z
UID:TALK172775@talks.cam.ac.uk
DESCRIPTION:W. Burnside proposed to characterize finite groups with a give
 n number of irreducible characters. The block-wise version of this problem
  is to characterize the defect groups of a $p$-block $B$ of a finite group
  $G$ with a given number of irreducible characters in it.In this context\,
  R. Brauer proved that $k(B)=1$ if\, and only if\, the defect group of $B$
  is trivial. Years later\, J. Brandt showed that $k(B)=2$ if\, and only if
 \, the defect groups of $B$ are cyclic of order 2. These two results do no
 t require the Classification of Finite Simple Groups. However\, the comple
 xity of this problem seems to explode when we deal with the next situation
 \, namely when we try to classify the defect groups of $p$-blocks satisfyi
 ng $k(B)=3$. It is conjectured 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 principal block\, then S.
  Koshitani and T. Sakurai have proven that $|D|=4$ or 5. We show that the 
 same is obtained whenever $B$ is an arbitrary block of $G$ and $D$ is norm
 al in $G$. Moreover\, we deal with the next natural situation\, namely whe
 re $k(B)=5$ and $B$ is the principal block of $G$\, in which case we obtai
 n that $D\\cong {\\sf C}_5\,{\\sf C}_7\, {\\sf D}_8\, {\\sf Q}_8$.This tal
 k is an overview of joint works with J.M. Mart&iacute\;nez\, L. Sanus\, M.
  Schaeffer Fry and C. Vallejo.
LOCATION:Seminar Room 1\, Newton Institute
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