The semidirect product of groups is a generalization of the direct product construction and provides a much larger collection of groups than one is able to obtain only by constructing new groups as direct products of groups that belong to the well-known standard group families.
This MSc thesis is a continuation of the study initiated by the author’s Bachelor’s thesis entitled Semidirect product of groups, which introduced the semidirect product of groups and provided various examples, mostly of semidirect products of cyclic groups. Some interesting results were proved, which however gave rise to several questions and possibilities for further research.
The MSc thesis investigates the relevance of the semidirect product of groups, mostly in group theory, but also in a wider context, e.g. in the field of discrete mathematics, where symmetries of combinatorial objects such as graphs are investigated. We are interested in the question how the knowledge on this construction complements the knowledge on groups, which is obtained only from direct products and basic standard group families. One of the most important questions that we investigate is how many groups, that cannot be constructed as direct products of smaller groups, can be obtained using semidirect products. On the other hand, we are also interested in the question of how many groups cannot be constructed using semidirect products. We investigate isomorphisms of semidirect products, where a cyclic group takes the place of the so-called complement of the normal group. We show that all groups of specific orders (e.g. the product of two prime numbers) are semidirect products. We use a concrete example to demonstrate how to find all groups of a given order if we know that all groups of this order are semidirect products. Finally, we discuss the wreath product, a special case of a semidirect product, and we give examples of where else it is possible to find these constructions beyond group theory, for example in graph theory.
The major objective of this thesis is to provide a detailed insight into the construction of the semidirect product and to find concrete examples of its application in both group theory and further, beyond the borders of abstract algebra. The semidirect product of groups is presented in a way suitable for those familiar with basic notions of group theory, yet still in need of an introduction to semidirect products.