We prove a broad generalization of a theorem of W. Burnside on real characters using permutation characters. Under a necessary hypothesis, we can give some control on multiplicities (a result that needs the Classification of Finite Simple Groups). Along the way, we also give a new characterization of the 2-closed finite groups using odd-order real elements of the group. All this can be seen as a contribution to Brauer's Problem 11. We also obtain similar results for 2-Brauer characters.

中文翻译：

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置换字符的实数成分

我们使用置换字符证明了 W. Burnside 定理对真实字符的广泛推广。在必要的假设下，我们可以对多重性进行一些控制（需要有限单群分类的结果）。在此过程中，我们还使用群的奇数阶实元素给出了 2-闭有限群的新特征。所有这些都可以看作是对 Brauer 问题 11 的贡献。我们也对 2-Brauer 字符获得了类似的结果。