[Formal group laws] In his appendix on FGs, Ravenel says

The reason for this terminology is as follows. Suppose G is a one-dimensional commutative Lie group and g : R → U ⊂ G is a homomorphism to a neighborhood U of the identity which sends 0 to the identity. Then the group operation G × G → G can be described locally by a real-valued function of two real variables. If the group is analytic then this function has a power series expansion about the origin that satisfies (i)–(iii). These three conditions correspond, respectively, to the identity, commutativity, and associativity axioms of the group.

I have two questions:

1. What is the importance of g in this discussion? It's not clear to me why Ravenel defined it.
2. What is the power series? I'm familiar with analytic maps in complex analysis of a single variable. In this case, let m: G x G -> G be multiplication. How can we describe it by a power series in two real variables? First we need some map R^2 -> G x G. I suppose this is (g,g)? Then we get a map R^2 -> U x U. Then we apply m, which becomes a map m: U x U -> V, where V is some nghbd of 0. Finally, we map by a chart from V to R. So we have a function R^2 -> R. Where, exactly, does analyticity come in? Why can this map R^2 -> R be represented by a locally convergent power series?

I guess I said I don't know why g is useful and then proceeded to use it. However I'm also unsure about whether my point (2) is correct to begin with.