Mutually Normalizing Regular Permutation Groups and Zappa-Szep Extensions of the Holomorph

Abstract

For a group G, embedded in its group of permutations B = P erm(G) via the left regular representation λ : G → P erm(G), the normalizer of λ(G) in B is Hol(G), the holomorph of G. It is known that Hol(G) is also the normalizer of ρ(G) as well and both λ(G) and ρ(G) are canonical examples of regular subgroups. The determination of those regular N ≤ P erm(G), where N ∼ = G with the same normalizer is keyed to the structure of the so-called multiple holomorph of G, NHol(G) = N orm B (N orm B (λ(G))). We wish to analyze the set of those subgroups of Hol(G) which mutually normalize each other, but which don't necessarily have the same normalizer. This analysis gives rise to a group that contains NHol(G), but is potentially larger. We introduce what we call the quasi-holomorph of G which allows one to enumerate this set of mutually normalizing subgroups of Hol(G). The quasi-holomorph will be a group properly containing Hol(G) and is frequently a Zappa-Szep product with the holomorph.