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Group rings of finite simple groups. (English) Zbl 1047.20007
Summary: Let \(G\) be a finite non-Abelian simple group. In the first part we consider the question whether \(\mathbb{C} G\) determines \(G\) up to isomorphism. This question is closely related to a recent conjecture of B. Huppert that \(G\) is determined up to a direct Abelian factor by its set of ordinary character degrees. We sketch a proof that a finite simple group is determined by all its group algebras over a field. This proof involves also arguments of modular group algebras of \(G\). The second part deals with conjugacy questions in the unit group of \(\mathbb{Z} G\). A survey of the known results of conjectures of Zassenhaus and variations of these conjectures is given with respect to finite simple groups.

MSC:
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
16S34 Group rings
16U60 Units, groups of units (associative rings and algebras)
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