\(p^ *\)-theory and modular representation theory
type: Item
instance of: scholarly article
title: \(p^ *\)-theory and modular representation theory
zbMATH Open document ID: 0618.20007
DOI: 10.1016/0021-8693(86)90243-7
author: Wolfgang Willems
published in: Journal of Algebra
publication date: 1986-01-01T00:00:00Z
review text: Let F denote a field of characteristic \(p>0\) and let G be a finite group. Let \(f_ 0(G)=\sum_{g\in G}a_ gg\) denote the primitive central idempotent of the principal b-block, \(B_ 0(G)\), of the group algebra FG. Set \(Supp(f_ 0(G))=\{g\in G|\) \(a_ g\neq 0\}\) and \(O_{f_ 0}(G)=<g|\) \(g\in Supp(f_ 0(G))>\), so that \(Supp(f_ 0(G))\) is a characteristic subset of G and \(O_{f_ 0}(G)\) is a characteristic subgroup of G. It is known that \(O_{f_ 0}(G)\) depends only on G and the characteristic p of F. This interesting paper demonstrates connections between \(O_{f_ 0}(F)\) and the generalized p'-core of G, \(O_ p*(G)\) [cf. \textit{H. Bender}, Hokkaido Math. J. 7, 271-288 (1978; Zbl 0405.20015)]. The main results of the paper are: Theorem 2.1: \(O_{f_ 0}(G)\) is a \(p^*\)-group. In particular, \(O_{f_ 0}(G)\leq O_ p*(G).\) Theorem 2.2: \(O_{f_ 0}(N_ G(P))=O_{f_ 0}(C_ G(P))\leq O_{f_ 0}(G)\) for all p-subgroups P of G. Theorem 2.6: If \(p\neq 2\), then \(O_{f_ 0}(G)=O_ p*(G).\) The paper concludes by describing the necessary alterations for the \(p=2\) case in Remark 2.8.
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20C05
zbMATH DE Number: 4002355
zbMATH Keywords: primitive central idempotent
zbMATH Keywords: principal b-block
zbMATH Keywords: group algebra
zbMATH Keywords: characteristic subgroup
zbMATH Keywords: generalized p'-core
zbMATH Keywords: \(p^ *\)-group
zbMATH Keywords: p-subgroups
reviewed by: Morton E. Harris
MaRDI profile type: Publication
full work available at URL: 0021-8693(86)90243-7
OpenAlex ID: W2086910857
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description: scientific article; zbMATH DE number 4002355