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

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 p³ 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 \(B\).

Name dc:type.qualificationname

d_ph

Level dc:type.qualificationlevel

d_ph

Grantor dc:publisher.institution

University of Birmingham

Year dc:date.issued

2004

Author dc:creator

• Alghamdi, Ahmad M.

• QA Mathematics