Computing normalisers of intransitive groups

TL;DR

This paper introduces a faster algorithm for computing normalisers of intransitive groups, significantly improving efficiency for groups with many orbits, and explores the problem's computational complexity.

Contribution

It presents a new, more efficient algorithm for normaliser computation in groups with many orbits and establishes the problem's complexity relation to automorphisms of linear codes.

Findings

New algorithm outperforms previous GAP implementations by many orders of magnitude.

Normaliser problem for G=S_n is as hard as computing monomial automorphisms of linear codes.

Provides complexity insights linking group normalisers to coding theory automorphisms.

Abstract

The normaliser problem takes as input subgroups $G$ and $H$ of the symmetric group $S_n$, and asks one to compute $N_G(H)$. The fastest known algorithm for this problem is simply exponential, whilst more efficient algorithms are known for restricted classes of groups. In this paper, we will focus on groups with many orbits. We give a new algorithm for the normaliser problem for these groups that performs many orders of magnitude faster than previous implementations in GAP. We also prove that the normaliser problem for the special case $G=S_n$ is at least as hard as computing the group of monomial automorphisms of a linear code over any field of fixed prime order.