The handout gives a much more detailed description of ``Free Products of Groups'' than Hatcher's book. Therefore, I want to take the next couple of days out of the handout. In this lecture, I want you to begin to cover section 4 (Chapter 3) of the handout called ``Free Products of Groups.'' You should not need to know sections 1-3 to be able to understand this section. You should state the definition of the free product on p.71 and prove they are unique (Proposition 4.1). In order to prove this, you will need to take the proof of Proposition 2.2 and modify it slightly (there are essentially the same proof). You can come and talk to me about the proof if you would like. We will prove the existence of this group in the next lecture. Before we can prove it, we will need some facts from Appendix B (handout), section 1. For this, you should recall the definition of the group of permutations of a set. Then you should state the definition of a left G-space and give an example. State and prove Theorem 1.1. Point out that this implies that there is a group homomorphism φ : G -> P(E)=permutations of E defined by φ(g)(x)=g . x for any x in E. State the definition of G acting effectively.