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

Joshua A. Grochow, Dan Johnson, Michael Levet

TL;DR

The paper advances the understanding of Group Isomorphism by solving isomorphism testing for Fitting-free groups in the parallel AC^3 model when groups are given by Cayley tables, using the fact that G/PKer(G) has small permutation degree $O(\log |G|)$. It also shows GI-hardness for Fitting-free groups specified by permutation generators via reductions from Graph Isomorphism and Linear Code Equivalence, highlighting a separation between input models. A key methodological thread is embedding G into automorphism groups of the socle and reducing the problem to Twisted Code Equivalence and Coset Intersection on small domains, enabling deep parallelization. Finally, the paper proves that Fitting-free groups are identifiable by first-order logic with only $O(\log\log n)$ variables, contrasting with Abelian families and connecting descriptive complexity to algebraic structure.

Abstract

In this paper, we exhibit an $\textsf{AC}^{3}$ 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 $G/\text{PKer}(G)$ 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 $\textsf{AC}^{3}$. 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 $\textsf{GI}$-hard and having no known subexponential-time algorithm). Lastly, we show that any Fitting-free group of order $n$ is identified by $\textsf{FO}$ 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 $\textsf{FO}$ formulas with $o(\log n)$ variables (Grochow & Levet, FCT '23).

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

TL;DR

The paper advances the understanding of Group Isomorphism by solving isomorphism testing for Fitting-free groups in the parallel AC^3 model when groups are given by Cayley tables, using the fact that G/PKer(G) has small permutation degree . It also shows GI-hardness for Fitting-free groups specified by permutation generators via reductions from Graph Isomorphism and Linear Code Equivalence, highlighting a separation between input models. A key methodological thread is embedding G into automorphism groups of the socle and reducing the problem to Twisted Code Equivalence and Coset Intersection on small domains, enabling deep parallelization. Finally, the paper proves that Fitting-free groups are identifiable by first-order logic with only variables, contrasting with Abelian families and connecting descriptive complexity to algebraic structure.

Abstract

In this paper, we exhibit an 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 (Babai, Codenotti, and Qiao; ICALP '12). Here, we leverage the fact that can be viewed as permutation group of degree . As is given by its multiplication table, we are able to implement the solution for the corresponding instance of Twisted Code Equivalence in . 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 -hard and having no known subexponential-time algorithm). Lastly, we show that any Fitting-free group of order is identified by formulas (without counting) using only variables. This is in contrast to the fact that there are infinite families of Abelian groups that are not identified by formulas with variables (Grochow & Levet, FCT '23).
Paper Structure (26 sections, 38 theorems, 28 equations)

This paper contains 26 sections, 38 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Groups and Codes
  4. Computational Complexity
  5. Permutation Group Algorithms
  6. Weisfeiler--Leman
  7. Pebbling Game
  8. Logics
  9. Coset Intersection for small-domain groups in parallel
  10. Twisted Code Equivalence over small domains in parallel
  11. Permutational Isomorphisms
  12. Permutational isomorphism of transitive permutation groups on small domains in parallel
  13. Additional preliminaries: primitivity and structure trees
  14. Parallel algorithms for small-domain structure trees and primitive groups
  15. Parallel algorithms for small-domain transitive permutation groups
  16. ...and 11 more sections

Key Result

Theorem 1.1

Isomorphism of Fitting-free groups given by their multiplication tables can be decided in $\mathsf{AC}^3$. More specifically, there is a uniform $\mathsf{AC}^3$ algorithm that, given two groups $G,H$ of order $n$, decides if both are Fitting-free, and if they are, correctly decides whether they are

Theorems & Definitions (79)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Proposition 3.1
  • proof
  • ...and 69 more