Counting characters in blocks

Abstract: Let $G$ be a finite group, let $p$ be a prime number and let $B$ be a $p$-block of $G$ with defect group $D$. Studying the structure of $D$ by means of the knowledge of some aspects of $B$ is a main area in character theory of finite groups. Let $k(B)$ be the number of irreducible characters in the $p$-block $B$. It is well-known that $k(B)=1$ if, and only if, $D$ is trivial. It is also true that $k(B)=2$ if, and only if, $|D|=2$. For blocks $B$ with $k(B)=3$ it is conjectured that $|D|=3$.

In this talk we restrict our attention to the principal $p$-block of $G$, $B_0(G)$, that is, the $p$-block containing the trivial character of $G$. In this case, by work of Belonogov, Koshitani and Sakurai we know the structure of $D$ when $k(B_0(G))=3$ or $4$. In this work, we go one step further and analyze the structure of D when $k(B_0(G))=5$.

This is a joint work with Mandi Schaeffer Fry and Carolina Vallejo.

group theory

Audience: researchers in the topic

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Finite Groups in Valencia

Series comments: The seminar meets weekly from February 26th to March 30th 2021.

For a detailed schedule and registration details, please visit the seminar webpage at

sites.google.com/view/finite-groups-seminar2021

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