Abstract

W. Burnside proposed to characterize finite groups with a given 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 not require the Classification of Finite Simple Groups. However, the complexity 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 satisfying $k(B)=3$. 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 normal in $G$. Moreover, we deal 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 an overview of joint works with J.M. Martínez, L. Sanus, M. Schaeffer Fry and C. Vallejo.