Abstract
In this paper, we study the structures of finite groups using some arithmetic conditions on the sizes of real conjugacy classes. We prove that a finite group is solvable if the prime graph on the real class sizes of the group is disconnected. Moreover, we show that if the sizes of all noncentral real conjugacy classes of a finite group G have the same 2-part and the Sylow 2-subgroup of G satisfies a certain condition, then G is solvable.
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