On short expressions for cosets of permutation subgroups

TL;DR

This paper demonstrates that permutation cosets, including all subgroups, can be described with short expressions within Babai's string isomorphism algorithm, providing insights into classification and explicit runtime constants.

Contribution

It establishes the feasibility of short expressions for permutation cosets in Babai's algorithm, both with and without CFSG, and explicates the recursion process with explicit constants.

Findings

Short expressions for permutation cosets are possible in Babai's algorithm.

The results hold both with and without CFSG assumptions.

Explicit constants for the algorithm's runtime are derived.

Abstract

Following Babai's algorithm for the string isomorphism problem, we determine that it is possible to write expressions of short length describing certain permutation cosets, including all permutation subgroups; this is feasible both in the original version of the algorithm and in its CFSG-free version, partially done by Babai and completed by Pyber. The existence of such descriptions gives a weak form of the Cameron-Mar\'oti classification even without assuming CFSG. We also thoroughly explicate Babai's recursion process (as given in Helfgott) and obtain explicit constants for the runtime of the algorithm, both with and without the use of CFSG.