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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.

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