Representing Quotients of Permutation Groups

IN this note, we consider the following problem. Let G be a finite permutation group of degree d, and let N b e a normal subgroup of G. Under what circumstances does G/N have a faithful permutation representation of degree at most di Positive answers to this question are likely to have applications to computational group theory, since there are currently no really satisfactory practical methods known for computing in quotients of permutation groups. Although Kantor and Luks demonstrate in [1] that essentially all of the algorithms that are known to have polynomial complexity for permutation groups G also have polynomial complexity for quotient groups G/N of G, they do not attempt to answer the above question, and it is not clear to what extent, if any, their proposed algorithms are practical. In view of this, it is somewhat surprising that the only answer to the question in the literature appears to be the negative result of P. M. Neumann in [2]. He observes that if we take G to be a direct product of n copies of the dihedral group D8 of order 8, with its natural permutation action of degree 4/i having n orbits of size 4, and then take N to be a central subgroup of G of order 2"" such that G/N is extraspecial, then the smallest faithful permutation representation of G/N has degree 2\ which can clearly be very much larger than 4n. More generally, we are likely to get negative answers of this type whenever G/N is some sort of central product, but one feature of these examples is that N is not a maximal normal subgroup of its class; specifically, it is not a maximal elementary abelian normal subgroup in Neumann's example. We shall prove two closely related positive results.