The normaliser problem has as input two subgroups H and K of the symmetric group $$\mathrm {S}_n$$
S
n
, and asks for a generating set for $$N_K(H)$$
N
K
(
H
)
: it is not known to have a subexponential time solution. It is proved in Roney-Dougal and Siccha (Bull Lond Math Soc 52(2):358–366, 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 $$\mathrm {S}_n$$
S
n
, in quasipolynomial time, we can decide whether $$N_{\mathrm {S}_n}(H)$$
N
S
n
(
H
)
is primitive, and if so, compute $$N_K(H)$$
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 in $$\mathrm {S}_n$$
S
n
is known not to be primitive.
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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.