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Finding Blocks of Imprimitivity When There is a Small-Base Action on Blocks

Robert Beals

TL;DR

This work extends efficient small-base primitivity testing to imprimitive groups that admit a small-base action on a block system by introducing a deep sifting framework that unifies transversal construction with base refinement. The authors achieve near-linear time $O(n\log^5 n)$ for most cases by leveraging the recent base-size bound $5\log n$ for nonlarge primitive groups (via the Kelsey–Roney-Dougal result) and a new variant of sifting, while providing sub-quadratic time for many large primitive actions through a refined base-size parameter $L$ and deep cube techniques. A certified nonredundant partial base and its certificate enable block discovery and a constructive path to imprimitivity, even when the action on blocks is not small-base. The results significantly broaden the range of groups amenable to sub-quadratic primitivity testing and block-finding, with practical implications for computational group theory and related applications. All mathematical expressions are presented with explicit $...$ delimiters to preserve precision across search and indexing systems.

Abstract

Given a transitive permutation group G of degree n , we seek to determine whether or not G is primitive, and to find a system of blocks of imprimitivity in the case that G is imprimitive. An algorithm of Atkinson solves this problem in time O(n^2) , while a previous algorithm of ours runs in time O(n log^3|G|) , which is advantageous in the small-base case. A simpler algorithm of Schonert and Seress has the same asymptotic O(n log^3|G|) performance. In this paper we extend the small-base algorithms to work with imprimitive groups G which, while not small-base in the action on n points, possess a small-base action on a block system. Using a recent upper bound by Kelsey and Roney-Dougal on the size of a nonredundant base of a primitive group of a given degree, we obtain a time of O(n log^5 n) except in the case that G has a primitive action (either on the n points or on a block system) for which the socle is isomorphic to Alt(m)^d for some m at least 5 and d at least 1. A key component of our improvement is a new variant of sifting, which is a workhorse of permutation group algorithms.

Finding Blocks of Imprimitivity When There is a Small-Base Action on Blocks

TL;DR

This work extends efficient small-base primitivity testing to imprimitive groups that admit a small-base action on a block system by introducing a deep sifting framework that unifies transversal construction with base refinement. The authors achieve near-linear time for most cases by leveraging the recent base-size bound for nonlarge primitive groups (via the Kelsey–Roney-Dougal result) and a new variant of sifting, while providing sub-quadratic time for many large primitive actions through a refined base-size parameter and deep cube techniques. A certified nonredundant partial base and its certificate enable block discovery and a constructive path to imprimitivity, even when the action on blocks is not small-base. The results significantly broaden the range of groups amenable to sub-quadratic primitivity testing and block-finding, with practical implications for computational group theory and related applications. All mathematical expressions are presented with explicit delimiters to preserve precision across search and indexing systems.

Abstract

Given a transitive permutation group G of degree n , we seek to determine whether or not G is primitive, and to find a system of blocks of imprimitivity in the case that G is imprimitive. An algorithm of Atkinson solves this problem in time O(n^2) , while a previous algorithm of ours runs in time O(n log^3|G|) , which is advantageous in the small-base case. A simpler algorithm of Schonert and Seress has the same asymptotic O(n log^3|G|) performance. In this paper we extend the small-base algorithms to work with imprimitive groups G which, while not small-base in the action on n points, possess a small-base action on a block system. Using a recent upper bound by Kelsey and Roney-Dougal on the size of a nonredundant base of a primitive group of a given degree, we obtain a time of O(n log^5 n) except in the case that G has a primitive action (either on the n points or on a block system) for which the socle is isomorphic to Alt(m)^d for some m at least 5 and d at least 1. A key component of our improvement is a new variant of sifting, which is a workhorse of permutation group algorithms.

Paper Structure

This paper contains 13 sections, 11 theorems, 6 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Transversals and sifting
  4. Base sizes of primitive groups
  5. Organization
  6. Deep Sifting
  7. Cubes
  8. Deep sifting
  9. Features of Deep Sifting
  10. Primitivity testing with deep sifting
  11. Beyond the small-base case
  12. Handling small-base primitive actions
  13. Some large cases in sub-quadratic time

Key Result

Theorem 1.1

Let $G\leq{\rm Sym}(\Omega)$ be a transitive group given by a set $S$ of generators. In time $O(n\log^2|G|\log n+n|S|\log |G|)$, we can determine whether or not $G$ is primitive, and in the imprimitive case we can find a block system.

Figures (1)

  • Figure 1: Pseudocode for deep sifting

Theorems & Definitions (15)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1: BabaiSzemeredi1984
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4: BabaiSzemeredi1984
  • Lemma 2.5
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • ...and 5 more