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

TL;DR

This paper extends small-base algorithms to identify blocks of imprimitivity in permutation groups with small-base actions on block systems, improving efficiency for certain classes of groups.

Contribution

It introduces a new algorithm that efficiently finds blocks of imprimitivity in groups with small-base actions on block systems, extending previous methods.

Findings

Achieves a time complexity of O(n log^5 n) for the problem.

Handles groups with small-base actions on block systems, not just on points.

Provides a new variant of sifting for permutation group algorithms.

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.