On finite groups with a certain Sylow normalizer. III
[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.
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