Semidirect Products : x - > ax + b as a First Example

After precalculus, mathematics tudents often leave behind the familiar family of transformations, x Fax + b. We will show that this family, when examined in the right light, leads us to some interesting and important ideas in group theory. By building on this accessible example, it is possible to introduce the semidirect product (a topic usually first seen at the graduate level) in an undergraduate abstract algebra course. After being introduced to the semidirect product, students are able to better understand the structures of some of the groups of small order that are typically discussed in a first course in abstract algebra.