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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)