Abstract
The idea of supercharacters for ordinary characters of a finite group G was introduced by Diaconis and Isaacs and further extended to Brauer characters by Chen and Lewis. The twin concepts of supercharacters and superclasses are further extended here to α-characters of G for α a complex-valued 2-cocycle of G. An α-quasi-supercharacter theory of G arises when the set of α-quasi-supercharacters of G are compatible with the set of α-regular quasi-superclasses of G. The structure of solvable groups that have exactly two α-quasi-supercharacter theories is determined.Communicated by Mark L. Lewis
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