The goal of this paper is to generalize several group theoretic concepts such as the center and commutator subgroup, central series, and ultimately nilpotence to a supercharacter theoretic setting, and to use these concepts to show that there can be a strong connection between the structure of a group and the structure of its supercharacter theories. We then use these concepts to show that the upper and lower annihilator series of $J$ can be described in terms of certain central series for the algebra group $G=1+J$ defined by $\mathsf{S}$, when $\mathsf{S}$ is the algebra group supercharacter theory defined by Diaconis--Isaacs.

中文翻译：

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超字符理论产生的幂零模拟

本文的目标是将中心和交换子子群、中心级数和最终幂零等几个群论概念推广到超字符理论设置，并使用这些概念表明一个群及其超性格理论的结构。然后我们使用这些概念证明 $J$ 的上下湮灭级数可以用 $\mathsf{S}$ 定义的代数群 $G=1+J$ 的某些中心级数来描述，当 $ \mathsf{S}$ 是由 Diaconis--Isaacs 定义的代数群超字符理论。