AbstractThe 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.