On the Descriptive Complexity of Groups without Abelian Normal Subgroups
🤖 AI Summary
This work investigates the logical description complexity of finite groups with no nontrivial abelian normal subgroups—i.e., direct products of nonabelian simple groups and their extensions.
Method: We introduce and analyze the binary Weisfeiler–Leman (2-WL) coloring algorithm for such groups, establishing its precise equivalence to the binary Ehrenfeucht–Fraïssé game. We further combine 2-WL with first-order logic augmented with generalized binary quantifiers.
Results: We prove that group isomorphism within this class is decided by 2-WL in a constant number of rounds using a constant number of pebbles. Consequently, these groups are definable in a fixed-variable, constant-quantifier-depth fragment of first-order logic—specifically, using O(1) variables and O(1) quantifier depth. This yields the first constant-depth logical characterization of nonabelian simple groups and their direct products, providing novel, low-complexity logical tools and theoretical foundations for group isomorphism testing.
📝 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.
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