Some Algorithms for Nilpotent Permutation Groups

LetG,HandEbe subgroups of a finite nilpotent permutation group of degreen. We describe the theory and implementation of an algorithm to compute the normalizerNG(H) in time polynomial inn, and we give a modified algorithm to determine whetherHandEare conjugate underGand, if so, to find a conjugating element ofG. Other algorithms produce the intersectionG ∩ Hand the centralizerCG(H). The underlying method uses the imprimitivity structure of G,H and an associated canonical chief series to reduce computation to linear operations. Implementations in GAP and Magma are practical for degrees large enough to present difficulties for general-purpose methods.