On the Descriptive Complexity of Groups without Abelian Normal Subgroups

TL;DR

The paper advances descriptive complexity for finite groups by formalizing a 2-ary Ehrenfeucht--Fraïssé framework and a corresponding 2-ary WL coloring that matches the q=2 game power. The authors prove a constant-bounded isomorphism-detection result for semisimple groups: a strategy using $7$ pebbles and $7$ rounds suffices to distinguish nonisomorphic groups, with the strength arising from identifying the socle $ ext{Soc}(G)$ and the conjugation action via $ ext{PKer}(G)$ and related invariants. This connects the game-theoretic distinction to FO logic with generalized $2$-ary quantifiers, offering a constructive view of isomorphism testing for a broad class of groups and clarifying the role of socle structure in descriptive complexity. The results suggest that, for semisimple groups, isomorphism testing can be captured by a fixed, small WL parameter, highlighting potential PTIME-like behavior in a nontrivial, natural class and motivating further exploration of hierarchy collapses and CFSG-independent approaches.

Abstract

In this paper, we explore the descriptive complexity theory of finite groups by examining the power of the second Ehrenfeucht--Fraïssé bijective pebble game in Hella's (Ann. Pure Appl. Log., 1989) hierarchy. This is a Spoiler--Duplicator game in which Spoiler can place up to two pebbles each round. While it trivially solves graph isomorphism, it may be nontrivial for finite groups, and other ternary relational structures. We first provide a novel generalization of Weisfeiler--Leman (WL) coloring, which we call 2-ary WL. We then show that 2-ary WL is equivalent to the second Ehrenfeucht--Fraïssé bijective pebble game in Hella's hierarchy. Our main result is that, in the pebble game characterization, only $O(1)$ pebbles and $O(1)$ rounds are sufficient to identify all groups without Abelian normal subgroups (a class of groups for which isomorphism testing is known to be in $\mathsf{P}$; Babai, Codenotti, & Qiao, ICALP 2012). We actually show that $7$ pebbles and $7$ rounds suffice. In particular, we show that within the first few rounds, Spoiler can force Duplicator to select an isomorphism between two such groups at each subsequent round. By Hella's results (ibid.), this is equivalent to saying that these groups are identified by formulas in first-order logic with generalized 2-ary quantifiers, using only $7$ variables and $7$ quantifier depth.

Theorems & Definitions (44)

• Theorem 1.1
• Definition 2.1
• Definition 2.2: Notation for pebbles, rounds, arity, and WL version
• Remark 2.3: Equivalence with logics with generalized $2$-ary quantifiers
• proof
• Definition 3.1: 2-ary $k$-dimensional Weisfeiler-Leman coloring
• Remark 3.2
• Theorem 3.3
• Theorem 3.3
• proof