Bounding p-Brauer characters in finite groups with two conjugacy classes of p-elements

TL;DR

This paper establishes bounds on the number of p-Brauer characters in the principal block of finite groups with specific conjugacy class structures, revealing exceptional cases and proposing a general lower bound related to group order and p-classes.

Contribution

It proves new bounds on p-Brauer characters in principal blocks for groups with two conjugacy classes of p-elements and proposes a general lower bound involving group order and p-classes.

Findings

If k(B_0)-l(B_0)=1, then l(B_0)  p-1 or p=11 with l(B_0)=9.

In groups where all non-trivial p-elements are conjugate, l(B_0) p-1 or p=11 with a specific structure.

A conjectured lower bound for the number of irreducible Brauer characters in principal blocks based on p and conjugacy classes.

Abstract

Let k(B_0) and l(B_0) respectively denote the number of ordinary and p-Brauer irreducible characters in the principal block B_0 of a finite group G. We prove that, if k(B_0)-l(B_0)=1, then l(B_0)\geq p-1 or else p=11 and l(B_0)=9. This follows from a more general result that for every finite group G in which all non-trivial p-elements are conjugate, l(B_0)\geq p-1 or else p = 11 and G/O_{p'}(G) =11^2:SL(2,5). These results are useful in the study of principal blocks with a few characters. We propose that, in every finite group G of order divisible by p, the number of irreducible Brauer characters in the principal p-block of G is always at least 2\sqrt{p-1}+1-k_p(G), where k_p(G) is the number of conjugacy classes of p-elements of G. This indeed is a consequence of the celebrated Alperin weight conjecture and known results on bounding the number of p-regular classes in finite groups.