On the Descriptive Complexity of Groups without Abelian Normal Subgroups (Extended Abstract)

TL;DR

This paper investigates the descriptive complexity of finite groups without Abelian normal subgroups, demonstrating that such groups can be distinguished using a limited number of pebbles and rounds in Ehrenfeucht-Fraisse games, with implications for logical definability.

Contribution

It introduces a novel 2-ary Weisfeiler-Leman coloring and shows its equivalence to a specific Ehrenfeucht-Fraisse game, establishing that these groups are identifiable with constant resources in logical formulas.

Findings

Groups without Abelian normal subgroups are identifiable with O(1) pebbles and rounds.

The 2-ary WL coloring is equivalent to the second Ehrenfeucht-Fraisse game.

Such groups are definable by first-order logic with generalized 2-ary quantifiers, using only O(1) variables and quantifier depth.

Abstract

In this paper, we explore the descriptive complexity theory of finite groups by examining the power of the second Ehrenfeucht-Fraisse 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 the 2-ary WL is equivalent to the second Ehrenfeucht-Fraisse 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 P; Babai, Codenotti, & Qiao, ICALP 2012). 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 O(1) variables and O(1) quantifier depth.