In this talk I will give a brief history of group theory along with some motivation as to why a person may study in this field. This development will be used to explain what a group extension is, as well as why someone might want to compute them.

Once we have motivated this area, I will talk about how I solve the problem of computing in group extensions. The solution depends upon doing computations using a subgraph of the Cayley Graph. In this development we will be using the concept of a Stalling Folding to prove that two graphs contain the same information. Once we prove that the information contained in both objects is the same, we will show how this can be used to express words as a product of conjugates of relators, and I will explain why this then solves the problem of computing in group extensions.

This talk with the assumes almost no background in group theory beyond what is covered in an introductory abstract algebra course.