On the Zappa-szép Product

The semidirect product of two groups is a natural generalization of the direct product of two groups in that the requirement that both factors be normal in the product is replaced by the weaker requirement that only one of the factors be normal in the product. The Zappa-Szép product of two groups is a natural generalization of the semidirect product of two groups in that neither factor is required to be normal. The Zappa-Szép product was developed by G. Zappa in [16]. Variations and generalizations were studied in the setting of groups by Rédei in [11] and Casadio in [3]. The products were used to discover properties of groups by Rédei, Szép and Tibiletti in numerous papers. Szép independently discovered the relations of the product and used them to study structural properties of groups (e.g., normal subgroups [13]) and also initiated the study of similar products in settings other than groups in [14] and [15]. The terminology Zappa-Szép product was suggested by Zappa. See also Page 674 of [8]. For the last several years, the author has been investigating and extending a family of closely related groups known as Thompson’s groups. See [2] for background on these groups. It was found that the structure of these groups as groups of fractions of monoids that in most cases were Zappa-Szép products of simpler monoids was the key to successful analysis. The author’s investigations were motivated in the first place by an intimate relationship that Thompson’s groups have with certain categories. The intimate relationship is helped by the fact that the Zappa-Szép product works fantastically well with categories. The (partial) multiplication on a cateogry is the composition of its morphisms. It turns out that the Zappa-Szép product has the remarkable property that it seems to require no hypotheses at all. Group-like properties such as associativity, fullness of the multiplication, identities, and inverses can be assumed or removed at random and the Zappa-Szép product can still be discussed and used at some level. This paper records the author’s observations on the Zappa-Szép product that are needed for his investigations. Properties needed to work with categories, monoids, and groups of fractions of monoids include associativity, existence of certain identities and inverses, cancellativity, and the existence of common right multiples. The behavior of the Zappa-Szép product in connection with these properties is studied in this paper. Definitions for these properties and the property in the next paragraph are given in the next section. Thompson’s groups are unusual in that they have strong finiteness properties (most are finitely presented) and some interesting geometric properties. The finiteness and geometric properties include the ability to act nicely on certain “nonpositively curved” metric spaces and spaces that are finite in each dimension modulo the action. Crucial to the construction of these spaces is the existence of least