Primitive normalisers in quasipolynomial time

TL;DR

This paper demonstrates that determining the primitivity of the normaliser in the symmetric group can be done in quasipolynomial time, reducing the general normaliser problem to the primitive case.

Contribution

It provides a quasipolynomial time algorithm to decide primitivity of the normaliser and compute it when primitive, simplifying the normaliser problem.

Findings

Decides normaliser primitivity in quasipolynomial time

Computes normaliser when it is primitive within quasipolynomial time

Reduces the general normaliser problem to the primitive case

Abstract

The normaliser problem has as input two subgroups $H$ and $K$ of the symmetric group $S_n$, and asks for a generating set for $N_K(H)$: it is not known to have a subexponential time solution. It is proved in [Roney-Dougal & Siccha, 2020] that if $H$ is primitive then the normaliser problem can be solved in quasipolynomial time. We show that for all subgroups $H$ and $K$ of $S_n$, in quasipolynomial time we can decide whether $N_{S_n}(H)$ is primitive, and if so compute $N_K(H)$. Hence we reduce the question of whether one can solve the normaliser problem in quasipolynomial time to the case where the normaliser is known not to be primitive.