On the Zappa-Szep Product

The Zappa-Szep product was developed independently by Guido Zappa and Jeno Szep as a generalization of the semidirect product: in the Zappa-Szep product, neither factor is required to be normal. We explore the basic properties of the product and show that it applies to settings more general than groups. The product is remarkable in that it requires almost no hypotheses to fuction at all and is adaptable to a very large number of situations.