Dade's conjecture for symmetric groups
type: Item
instance of: scholarly article
title: Dade's conjecture for symmetric groups
zbMATH Open document ID: 0839.20018
author: Katsuhiro Uno
author: Jørn B. Olsson
published in: Journal of Algebra
publication date: 1996-06-10T00:00:00Z
review text: Let \(G\) be a finite group and \(p\) be a prime number. A chain of \(p\)-subgroups \({\mathcal C}: P_0 \subset P_1 \subset\dots \subset P_m\) of \(G\) is called a radical chain if \(P_0= O_p (G)\) and \(P_i= O_p (\bigcap^i_{j=0} N_G (P_j))\) for \(i=1, 2, \dots, m\). The normalizer of the above chain is defined by \(N_G^{({\mathcal C})}= \bigcap^m_{j=0} N_G (P_j)\). Obviously \(G\) acts on the set of all the radical chains in \(G\) by conjugation. A set of representatives for the \(G\)-orbits is denoted by \({\mathcal R}/G\). For any subgroup \(H\) of \(G\), any \(p\)-block \(B\) of \(G\) and any non-negative integer \(d\) let \(k(H, B, d)\) denote the number of irreducible characters \(\varphi\) of \(H\) satisfying (i) \(\varphi\) belongs to a \(p\)-block \(b\) of \(H\) such that \(b^G =B\), (ii) The \(p\)-part of \(|H|/ \varphi(1)\) is equal to \(p^d\). A conjecture of Dade, called ordinary conjecture, states that if \(O_p (G)=1\) and \(B\) is a \(p\)-block of \(G\) with positive block, then for all \(d\geq 0\) the equality \(\sum_{c\in {\mathcal R}/G} (-1)^{|{cal C}|} k (N_G({\mathcal C}), B,d)=0\) holds. This conjecture has been verified for several simple groups. According to the authors' announcement the conjecture has been verified for any positive defect block of the group \(\text{GL}(n,q)\) in the defining characteristic. In the present paper the authors verify the conjecture of Dade for the symmetric groups in odd characteristic, i.e. when \(p\) is an odd prime.
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20D20
zbMATH DE Number: 815924
zbMATH Keywords: finite groups
zbMATH Keywords: chain of \(p\)-subgroups
zbMATH Keywords: radical chains
zbMATH Keywords: number of irreducible characters
zbMATH Keywords: \(p\)-blocks
zbMATH Keywords: ordinary conjecture
zbMATH Keywords: defect groups
zbMATH Keywords: symmetric groups
reviewed by: M. R. Darafsheh
MaRDI profile type: Publication
full work available at URL: jabr.1995.1258
OpenAlex ID: W1985209571
Wikidata QID: Q123272307
DOI: 10.1006/JABR.1995.1258
description: scientific article; zbMATH DE number 815924