On finite groups with a certain Sylow normalizer. III
type: Item
instance of: scholarly article
title: On finite groups with a certain Sylow normalizer. III
zbMATH Open document ID: 0283.20005
DOI: 10.1016/0021-8693(74)90084-2
author: Stephen D. Smith
published in: Journal of Algebra
publication date: 1974-01-01T00:00:00Z
review text: [Parts I and II, ibid. 26, 343--365 (1973; Zbl 0264.20012) and 26, 366--367 (1973; Zbl 0264.20013).] The paper completes the proof of the following: Theorem. Let \(G\) be a finite group, \(P\) a Sylow \(p\)-subgroup of \(G\), \(p\) odd. Suppose (1) \(\vert N(P)/P . C(P)\vert = 2\). (2) \(P\) has nilpotence class at most 2. Then either \(O^p(G)<G\) or \(O^2(G)<G\). In case \(O^p(G)=G\), we have in fact \(G =O_{p'}(G) . N(P)\). In view of the two previous papers in the series, the only case remaining to be proved is the particular case of assumption (2) given by: (2') \(P\) has nilpotence class 2, with \(\vert P'\vert >p\). Techniques involve consideration of the irreducible characters of the principal \(p\)-block of \(G\). The relevant analogue of ``exceptional character theory'' is required. It is interesting that the methods can be applied to the irreducible characters of an arbitrary Sylow \(p\)-group of class 2.
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20C25
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20J99
zbMATH DE Number: 3443817
MaRDI profile type: Publication
full work available at URL: 0021-8693(74)90084-2
OpenAlex ID: W4206824323
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cites work: On finite groups with a certain Sylow normalizer. I
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description: scientific article; zbMATH DE number 3443817