A Systematic Study of Isomorphism Invariants of Finite Groups via the Weisfeiler-Leman Dimension

TL;DR

This paper explores how the Weisfeiler-Leman algorithm can effectively identify and compare various classical invariants of finite groups, revealing its power in understanding group isomorphism and structure.

Contribution

It demonstrates that a low-dimensional Weisfeiler-Leman algorithm can detect many key group invariants and provides a new canonical decomposition tool, advancing group isomorphism analysis.

Findings

WL algorithm detects centers, automorphism groups, and derived series.

It determines isomorphism types of socles and composition factors.

WL dimension increases by at most 1 under direct product decomposition.

Abstract

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.