We generalize the common notion of descending and ascending central series. The descending approach determines a naturally graded Lie ring and the ascending version determines a graded module for this ring. We also link derivations of these rings to the automorphisms of a group. This process uncovers new structure in 4/5 of the approximately 11.8 million groups of size at most 1000 and beyond that point pertains to at least a positive logarithmic proportion of all finite groups.

中文翻译：

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群的新 Lie 积及其自同构

我们概括了降序和升序中央序列的常见概念。降序方法确定自然分级的 Lie 环，升序版本确定该环的分级模块。我们还将这些环的推导与群的自同构联系起来。这个过程揭示了大约 1180 万个组中的 4/5 的新结构，最多 1000 个，并且超过该点至少与所有有限组的正对数比例有关。