Conjectures on character degrees for the simple Thompson group.
type: Item
instance of: scholarly article
title: Conjectures on character degrees for the simple Thompson group.
zbMATH Open document ID: 1057.20005
author: Katsuhiro Uno
published in: Osaka Journal of Mathematics
publication date: 2004-05-27T00:00:00Z
review text: Over the past 20 years or so, a whole collection of conjectures has been developed, relating the character degrees of finite groups to the character degrees of various \(p\)-local subgroups. The simplest version, due to McKay, is that the number of characters of \(G\) of degree prime to \(p\) is equal to the same number for the Sylow \(p\)-normalizer. Alperin extended this to a block version, relating character degrees of blocks with defect group \(D\) to the character degrees of \(N_G(D)\). Further refinements due to Dade extend this further by considering the numbers of characters in each \(p\)-block whose degrees are divisible by any given power of \(p\). The Dade conjectures are expressed in terms of an alternating sum of these numbers, summed over the normalizers of the so-called `radical \(p\)-chains'. In a different direction, Isaacs and Navarro refine the Alperin-McKay conjecture by considering the congruences modulo \(p\) of the \(p'\)-parts of the character degrees. In this paper, Uno proposes a marriage of the two, known by other authors as Uno's conjecture. Essentially, Dade's conjecture is split into a sum of \((p-1)/2\) conjectures, one for each pair \(\{\pm k\}\) of non-zero values modulo \(p\). The bulk of the present paper is devoted to a verification of this new conjecture for the sporadic simple Thompson group.
reviewed by: Robert A. Wilson
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20C34
zbMATH DE Number: 2068624
zbMATH Keywords: character degrees
zbMATH Keywords: Dade conjecture
zbMATH Keywords: sporadic simple groups
zbMATH Keywords: numbers of characters
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2068624