Abstract
The group ring of a finite group G over a field K is a symmetric algebra and hence an injective ring. If K is of characteristic zero, KG is semisimple. Thus the center of KG is injective. We proof in this paper: If K is of characteristic p>0, then the center of KG is injective, if and only if G is p-nilpotent and the p-Sylow-subgroups of G are abelian. We determine also the structure of these group rings. This will be done more generally for symmetric algebras which have an injective center and split over its Jacobson-radical.
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Müller, W. Symmetrische Algebren mit injektivem Zentrum. Manuscripta Math 11, 283–289 (1974). https://doi.org/10.1007/BF01173719
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DOI: https://doi.org/10.1007/BF01173719