We investigate the relationship between various isomorphism invariants for finite groups. Specifically, we use the Weisfeiler-Leman dimension (WL) to characterize, compare and quantify the effectiveness and complexity of invariants for group isomorphism. It turns out that a surprising number of invariants and characteristic subgroups that are classic to group theory can be detected and identified by a low dimensional Weisfeiler-Leman algorithm. These include the center, the inner automorphism group, the commutator subgroup and the derived series, the abelian radical, the solvable radical, the Fitting group and $\pi$-radicals. A low dimensional WL algorithm additionally determines the isomorphism type of the socle as well as the factors in the derived series and the upper and lower central series. We also analyze the behavior of the WL algorithm for group extensions and prove that a low dimensional WL algorithm determines the isomorphism types of the composition factors of a group. Finally we develop a new tool to define a canonical maximal central decomposition for groups. This allows us to show that the Weisfeiler-Leman dimension of a group is at most one larger than the dimensions of its direct indecomposable factors. In other words the Weisfeiler-Leman dimension increases by at most 1 when taking direct products.
翻译:具体地说, 我们使用 Weisfeiler- Leman 维基、 可溶性激进、 配配组和 $\\pi$- radics 来描述、 比较和量化该组异异异异质的效能和复杂性。 事实证明, 典型为群数理论的变异性和特征分组数量惊人, 可以被一个低维 Weisfeiler- Leman 算法所检测和识别。 其中包括中心、 内自闭式组、 通量分组和衍生序列、 ABelian 激进、 可溶性激进、 配装组和 $\pi$- radicals 。 低维维基 WL 算法还分析了WL 算法对于群扩展的行为, 并证明低维维维维的算法决定了一个组的组成因素的不形态类型。 最后, 我们开发了一个新的工具, 来定义该组最大型组群群的中最大核心脱composition 。 这让我们在最直接的值上显示“ Wedeal- Ledeal ” 时, 我们在最直接的一组中可以显示“ 。