Thinking of groups When I think of a group, I can't help but thinking of a function...But those are two different things.
How can I ''view'' a group? For example, what kind of group could the natural log be?
A group consisting of the set of positive reals under some operation? What kind of relations would be in this group?
Or am I thinking it backward?
 A: All the data of a group $G$ is encoded in the function
$$\mu : G\times G\rightarrow G$$
Which should be thought of as the group operation, and which satisfies the group axioms. In this way, a group can be thought of as a function.
You might also be interested in reading about formal groups, which is essentially just the group operation without a set. (see the wikipedia article).
I don't see any way of seeing the natural log as a group. However, you can think of it as a group homomorphism from the multiplicative group $\mathbb{R}^\times$ to the additive group $\mathbb{R}^+$. This follows from the simple identity $\ln ab = \ln a + \ln b$.
If you want to think of the set of the positive reals under an operation, as a group the natural operation would be multiplication.
A: You can, however, see a group as a set of functions. More precisely, as a set of permutations (recall that a permutation is a bijection of a set onto itself). Indeed, if $G$ is a group with operation $\cdot$, an element $a$ of $G$ can be viewed as the function from $G$ to $G$ which sends $x$ to $a\cdot x$ (show that it is a bijection!). This is in particular why the element $e$ such that $e\cdot x = x\cdot e = x$ for all $x$ is sometimes called the identity, because when you view it as a function, it is the identity function.
