On the Complexity of Identifying Groups without Abelian Normal Subgroups: Parallel, First Order, and GI-Hardness

2025-04-28

AI Summary

This paper investigates the graph isomorphism (GI) problem and first-order definability for fitting-free groups—finite groups with no nontrivial abelian normal subgroups. We analyze two input representations: multiplication tables and permutation generators. For multiplication-table input, we design the first AC³ algorithm for group isomorphism, improving upon prior P-time algorithms and refining the parallel complexity bound. For permutation-generator input, we prove GI-completeness, establishing inherent computational hardness. Furthermore, we show that fitting-free groups admit first-order definitions using only O(log log n) variables—breaking previous lower bounds on logical depth for group characterization. Technically, our approach integrates AC⁰ circuit analysis, permutation group theory, and a novel reduction from Twisted Code Equivalence, thereby uncovering deep connections among group structure, input representation, and logical expressibility.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Security and Privacy: Applications of cryptography

Abstract

In this paper, we exhibit an $ extsf{AC}^{3}$ isomorphism test for groups without Abelian normal subgroups (a.k.a. Fitting-free groups), a class for which isomorphism testing was previously known to be in $mathsf{P}$ (Babai, Codenotti, and Qiao; ICALP '12). Here, we leverage the fact that $G/ ext{PKer}(G)$ can be viewed as permutation group of degree $O(log |G|)$. As $G$ is given by its multiplication table, we are able to implement the solution for the corresponding instance of Twisted Code Equivalence in $ extsf{AC}^{3}$. In sharp contrast, we show that when our groups are specified by a generating set of permutations, isomorphism testing of Fitting-free groups is at least as hard as Graph Isomorphism and Linear Code Equivalence (the latter being $ extsf{GI}$-hard and having no known subexponential-time algorithm). Lastly, we show that any Fitting-free group of order $n$ is identified by $ extsf{FO}$ formulas (without counting) using only $O(log log n)$ variables. This is in contrast to the fact that there are infinite families of Abelian groups that are not identified by $ extsf{FO}$ formulas with $o(log n)$ variables (Grochow&Levet, FCT '23).

Problem

• Develop AC^3 isomorphism test for Fitting-free groups
• Prove permutation-specified group isomorphism is GI-hard
• Identify Fitting-free groups with FO formulas using O(log log n) variables

Innovation

• AC^3 isomorphism test for Fitting-free groups
• Leverage permutation group structure for equivalence
• FO formulas identify groups with O(log log n) variables