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

Alghamdi, Ahmad M. (2004)
Ph.D. thesis, University of Birmingham.

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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 \(B\).

Type of Work:Ph.D. thesis.
Supervisor(s):Robinson, Geoffrey R.
School/Faculty:Schools (1998 to 2008) > School of Mathematics & Statistics
Department:Mathematics
Keywords:Representation of Finite Groups; Block Theory, Alperin's Conjecture, Extra Special Groups
Subjects:QA Mathematics
Institution:University of Birmingham
Library Catalogue:Check for printed version of this thesis
ID Code:86
This unpublished thesis/dissertation is copyright of the author and/or third parties. The intellectual property rights of the author or third parties in respect of this work are as defined by The Copyright Designs and Patents Act 1988 or as modified by any successor legislation. Any use made of information contained in this thesis/dissertation must be in accordance with that legislation and must be properly acknowledged. Further distribution or reproduction in any format is prohibited without the permission of the copyright holder.
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