The Ordinary Weight conjecture and Dade's Projective Conjecture for p-blocks with an extra-special defect group

Alghamdi, Ahmad M. (2004). The Ordinary Weight conjecture and Dade's Projective Conjecture for p-blocks with an extra-special defect group. University of Birmingham. Ph.D.

Preview

Alghamdi04PhD.pdf
PDF
Download (580kB)

Abstract

Let $p$ be a rational odd prime number, $G$ be a finite group such that $|G|=p^am$ , with $p \nmid m$ . Let $B$ be a $p$ -block of $G$ with a defect group $E$ which is an extra-special $p$ -group of order $p^3$ and exponent $p$ . Consider a fixed maximal $(G, B)$ -subpair $(E, b_E)$ . Let $b$ be the Brauer correspondent of $B$ for $N_G(E, b_E)$ . For a non-negative integer $d$ , let $k_d(B)$ denote the number of irreducible characters $\chi$ in $B$ which have $\chi(1)_p=p^{a-d}$ and let $k_d(b)$ be the corresponding number of $b$ . Various generalizations of Alperin's Weight Conjecture and McKay's Conjecture are due to Reinhard Knorr, Geoffrey R. Robinson and Everett C. Dade. We follow Geoffrey R. Robinson's approach to consider the Ordinary Weight Conjecture, and Dade's Projective Conjecture. The general question is whether it follows from either of the latter two conjectures that $k_d(B)=k_d(b)$ for all $d$ for the $p$ -block $B$ . The objective of this thesis is to show that these conjectures predict that $k_d(B)=k_d(b)$ , for all non-negative integers $d$ . It is well known that $N_G(E, b_E)/EC_G(E)$ is a $p^'$ -subgroup of the automorphism group of $E$ . Hence, we have considered some special cases of the above question.The unique largest normal $p$ -subgroup of $G$ , $O_p(G)$ is the central focus of our attention. We consider the case that $O_p(G)$ is a central $p$ -subgroup of $G$ , as well as the case that $O_p(G)$ is not central. In both cases, the common factor is that $O_p(G)$ is strictly contained in the defect group of $B$ .