Bounding $p$-regular conjugacy classes and $p$-Brauer characters in finite groups

Abstract: We discuss two closely related problems on bounding the number of $p$-regular conjugacy classes of a finite group $G$ and bounding the number of irreducible $p$-Brauer characters of $G$ or a block of $G$. Among other results we will show that the number of $p$-regular classes of a finite group $G$ is bounded below by $2\sqrt{p−1}+1−k_p(G)$, where $k_p(G)$ is the number of classes of $p$-elements of $G$. This and the celebrated Alperin weight conjecture imply the same bound for the number of irreducible $p$-Brauer characters in the principal $p$-block of $G$. We also discuss the bounds in the minimal situation when $G$ has a unique class of nontrivial $p$-elements, which have applications to the study of principal blocks with few characters. The talk is based on joint works with A. Moretó, with A. Maroti, and with B. Sambale and P.H. Tiep.

group theory

Audience: researchers in the topic

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