Abstract
The normaliser problem has as input two subgroups H and K of the symmetric group mathrm {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 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, in quasipolynomial time, we can decide whether N_{mathrm {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 in mathrm {S}_n is known not to be primitive.
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