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On the torsion units of some integral group rings. (English) Zbl 1097.16009

Theorem: If the finite group \(G\) has a normal Sylow-\(p\) subgroup with Abelian complement, then each torsion unit in \(\mathbb{Z}_{(p)}G\) is rationally conjugate to a trivial unit. – The proof uses A. Weiss’ papers on the Zassenhaus conjecture for nilpotent groups [Ann. Math. (2) 127, No. 2, 317-332 (1988; Zbl 0647.20007); J. Reine Angew. Math. 415, 175-187 (1991; Zbl 0744.16019)]; it also relies on new information about the partial augmentation of torsion units of prime power order, for example: if \(p\not\in R^\times\) (where \(R\) is a Dedekind domain of characteristic \(0\)), then \(\varepsilon_g(u)=0\) for every torsion unit \(u\in RG\) of order \(p^n\) (and augmentation 1) and every \(g\in G\) whose \(p\)-part has order \(>p^n\).
The theorem covers lots of known special results on torsion units in group rings [see, e.g., S. K. Sehgal’s book ‘Units in integral group rings’ (Longman, 1993; Zbl 0803.16022); M. A. Dokuchaev and S. O. Juriaans, Can. J. Math. 48, No. 6, 1170-1179 (1996; Zbl 0870.16020)].

MSC:

16U60 Units, groups of units (associative rings and algebras)
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
16S34 Group rings
20C11 \(p\)-adic representations of finite groups
20D15 Finite nilpotent groups, \(p\)-groups
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