ABSTRACT

Chapter 9. W hen Is a Group a Group? (Cayley’s T heorem)

We have just constructed several new candidates for groups, e.g., certain semidirect products and the quaternion group, and have shown by various ad hoc arguments in the exercises that they are indeed groups. Several of these proofs involved displaying the proposed group as a subgroup of Sn (cf. Exercises 8.3 and 8.6). We would like to see why these proofs work, and in doing so shall develop a general procedure that is guaranteed to work for any group.